|
|
| Line 1: |
Line 1: |
| {{about|an algebraic structure|geometric rings|Annulus (mathematics)|the set theory concept|Ring of sets}}
| |
| [[File:Chapitel IX. of Die Theorie der algebraischen Zahlkörper.png|thumb|Chapter IX of [[David Hilbert]]'s ''Die Theorie der algebraischen Zahlkörper.'' The chapter title is Die Zahlringe des Körpers, literally "the number rings of the field". The word "ring" is the contraction of "Zahlring".]]
| |
|
| |
|
| In [[mathematics]], and more specifically in [[algebra]], a '''ring''' is an [[algebraic structure]] with operations generalizing the [[arithmetic operation]]s of [[addition]] and [[multiplication]]. By means of this generalization, theorems from [[arithmetic]] are extended to non-numerical objects like [[polynomial]]s, [[series (mathematics)|series]], [[matrix (mathematics)|matrices]] and [[function (mathematics)|functions]].
| |
|
| |
|
| Rings were first formalized as a common generalization of [[Dedekind domain]]s that occur in [[number theory]], and of [[polynomial ring]]s and rings of invariants that occur in [[algebraic geometry]] and [[invariant theory]]. They are also used in other branches of mathematics such as [[geometry]] and [[mathematical analysis]]. The formal definition of rings is relatively recent, dating from the 1920s.
| | Aging adults are especially vulnerable to nutritious deficits, which means weight loss plans that do not supply their bodies together with the important factors required for ideal overall health. By focusing on food which are fast and simple to prepare, suitable diet is frequently diminished. This article offers valuable techniques so you can balance your daily diet inside a healthier fashion.<br><br>Keeping a consistent and [http://Anunturi.Brasovpress.ro/muscle-development-ideas-to-support-everybody-out/ effectively functioning] digestive system may help increase weight loss minimizing body mass total. Beverage all the water since you can on a daily basis.<br><br>Search for more techniques for getting far more health proteins in your daily diet than simply ingesting various meats every day. There are several other types of health proteins. These include nuts, tofu, beans, and sea food. You may decide to use some of these items as being a primary recipe, or incorporate them into other quality recipes. Involve numerous different varieties of protein in what you eat to stop you from getting bored using the same types of foods.<br><br>Making use of floor poultry as opposed to soil meat provides numerous benefits however, people typically dislike the dryness. As a way to remove this dry skin, you need to devote some EVOO (additional virgin organic olive oil) to create the poultry more juicy. If you loved this write-up and you would like to receive extra facts about [http://hellono.com/board/board_SfQK47/280859 Hellono.Com] kindly stop by our own web site. You can expect to eat less body fat and still obtain a delicious flavour inside your meats.<br><br>Choose toned soil turkey bust to use as a terrain beef substitute inside your food. Accomplishing this indicates you are having fewer calories and unhealthy fat. Just be certain you obtain terrain turkey chest darker turkey meat is a lot like meat, as far as nutritional value moves. Particular soil turkey has the two darker meat and chest, significance there may be nonetheless saturated fat.<br><br>The ideal way to prepare beef from the nutritional perspective are cooking, broiling, grilling, and roasting. Attempt preparing food mist rather than butter. Be sure to make sure you pressure your browned beef and rinse off with warm water. You could make lots of meals more healthy like that.<br><br>You are able to increase the dietary articles of cooked goods by replacing whole wheat grains flour for refined bright white flour. Whole wheat flour contains much more nutrients and vitamins and dietary fiber, and possesses a further benefit of being significantly less highly processed than white-colored flour.<br><br>Spice up your normal food choices by adding seaweed in your grocery list. The most popular variants--dulse, kombu and nori--contain a bevy of minerals and vitamins. Seashore residential areas have long integrated these types of plant life inside their diet programs.<br><br>Consider building a [http://Erotik-kontaktmark.com/author/ed6007/ veggie pizzas] that your whole loved ones will adore. As soon as you layer the cheese and pepperoni, add-on wholesome selections including olives, tomato plants, onions and peppers. Be sure they consume all of it.<br><br>This will likely keep you from venturing out with the very last minute and buying junk food. You will find a assortment of dish concepts all set at the moment's observe, so monotony is no problem.<br><br>A fantastic nutritious meal to serve is kabobs. This really is a class hard work since everyone can select what goes on their skewers. Create the hues brilliant and happy to make sure they would want to make theirs as quite (and veggie filled) as possible.<br><br>Prevent snacks rich in levels of fatty foods. Avoid food items with veggie oils and red meats. This commonly contains oils that will have even higher degrees of unhealthy fat as compared to animal goods. Saturated fat will be the bad kind of body fat which leads to loading on excess pounds. Even products that say they are doing not have cholesterol can raise your cholesterol levels.<br><br>Refreshing beets are great for you, but canned beets aren't. There are actually an excessive amount of sea salt in processed beets, new types are a far greater decision. Make an effort to steam your beet plants or incorporate beetroot within a salad.<br><br>Pantothenic acidity is a vital nutrient. Your system demands it for proper metabolism. Panthothenic acid solution also helps with the activity of nutrients. Try to eat beef and grain to acquire sufficient pantothenic acid solution.<br><br>When hunger attacks, ease the beast with a nourishing basic of vegetables and fruits. Comply with using a higher-health proteins entree and conclusion with carbohydrates should you need to. Although carbs are important in every diet regime, lots of people eat as well the majority of them. By eating lots of veggies, fruits and proteins, you won't have just as much place with your belly for too many carbs.<br><br>Make sure you consume adequate levels of food items. Lots of people are committed to not overeat, however, not consuming sufficient is equally as unhealthy. Beneath-consuming falls your blood sugar and causes you to susceptible to desires. It's advisable to have 5 - 6 small foods daily to suppress this.<br><br>If one makes a practice from learning new dishes typically, you can actually work in the direction of more healthy food items and getting good eating routine. Paying time needed to understand new dishes is actually a excellent approach to breakup the monotony at times connected with healthy having. Also, it is a way to make wholesome consuming fun and bold as opposed to boring and obligatory.<br><br>When having, your ought to prioritize vegatables and fruits, then health proteins. Carbohydrates should be your lowest concern. Despite the fact that sugars are essential, many people often take in an excessive amount of. Consuming fruit and vegetables, many fruits and protein at the start will give you less yearnings for big amounts of carbs.<br><br>Corn, legumes and chicken breast can be quite a basic, healthy meal. This food and similar ones give you a healthy, healthful blend of starches and proteins. This meal isn't particularly wonderful with regards to micronutrients, but overall, it really is a pretty wholesome food. Our company is almost certainly quite familiar with the meal pyramid that allows us know all we must eat. Even so, despite the fact that it's not improper, it's continue to very out-of-date because it doesn't consider any current scientific findings.<br><br>If you are picking healthy foods to enhance your diet program, keep in mind that foods contain a lot more diet natural than they do prepared. These food types are packed with nutrition and you should not drop them throughout cooking food. This truly retains correct for both fruits and veggies, so try to eat all these with their normal form.<br><br>Clearly, the foods we try to eat use a bearing on the quality of life we expertise. Use these ways to try to eat and are living a good daily life. Control your diet program and feel happy! |
| | |
| Briefly, a ring is an [[abelian group]] with a second [[binary operation]] that is distributive over the abelian group operation and is associative. The abelian group operation is called "addition" and the second binary operation is called "multiplication" in analogy with the [[integer]]s. One familiar example of a ring is the set of integers. The integers are a [[commutative ring]], since ''a'' times ''b'' is equal to ''b'' times ''a''. The set of polynomials also forms a commutative ring. An example of a non-commutative ring is the ring of [[square matrix|square matrices]] of the same size. Finally, a [[field (mathematics)|field]] is a commutative ring in which one can [[Divisibility (ring theory)|divide]] by any nonzero element: an example is the field of [[real number]]s.
| |
| | |
| Whether a ring is commutative or not has profound implication in the study of rings as abstract objects, the field called the [[ring theory]]. The development of the commutative theory, commonly known as [[commutative algebra]], has been greatly influenced by problems and ideas occurring naturally in [[algebraic number theory]] and [[algebraic geometry]]: important commutative rings include fields, polynomial rings, the [[coordinate ring]] of an [[affine algebraic variety]], and the [[ring of integers]] of a number field. On the other hand, the noncommutative theory takes examples from [[representation theory]] ([[group ring]]s), [[functional analysis]] ([[operator algebra]]s) and the theory of [[differential operators]] ([[ring of differential operators|rings of differential operators]]), and the topology ([[cohomology ring]] of a [[topological space]].)
| |
| | |
| {{Algebraic structures |Ring}}
| |
| | |
| == Definition and illustration ==
| |
| | |
| The most familiar example of a ring is the set of all [[integer]]s, '''Z''', consisting of the [[number]]s
| |
| | |
| :. . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . .
| |
| | |
| The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings.
| |
| | |
| ===Definition===
| |
| A '''ring''' is a [[Set (mathematics)|set]] ''R'' equipped with [[binary operation]]s<ref>Implicit in the assumption that + is a binary operation is that ''R'' is '''closed''' under +, meaning that for any ''a'' and ''b'' in ''R'', the value of ''a'' + ''b'' is defined to be an element of ''R''. The same applies to multiplication. Closure would be an axiom, however, only if, instead of binary operations on ''R'', we had functions + and · ''a priori'' taking values in some larger set ''S''.</ref> + and '''·''' satisfying the following eight axioms, called the '''ring axioms''':
| |
| | |
| * ''R'' is an [[abelian group]] under addition, meaning:
| |
| :1. (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'') for all ''a'', ''b'', ''c'' in ''R'' (+ is [[associativity|associative]]).
| |
| :2. There is an element 0 in ''R'' such that ''a'' + 0 = ''a'' and 0 + ''a'' = ''a'' (0 is the [[additive identity]]).
| |
| :3. For each ''a'' in ''R'' there exists −''a'' in ''R'' such that ''a'' + (−''a'') = (−''a'') + ''a'' = 0 (−''a'' is the [[additive inverse]] of ''a'').
| |
| :4. ''a'' + ''b'' = ''b'' + ''a'' for all ''a'' and ''b'' in ''R'' (+ is [[commutativity|commutative]]).
| |
| * ''R'' is a [[monoid]] under multiplication, meaning:
| |
| :5. (''a'' ⋅ ''b'') ⋅ ''c'' = ''a'' ⋅ (''b'' ⋅ ''c'') for all ''a'', ''b'', ''c'' in ''R'' (⋅ is associative).
| |
| :6. There is an element 1 in R such that ''a'' ⋅ 1 = ''a'' and 1 ⋅ ''a'' = ''a'' (1 is the [[multiplicative identity]]).
| |
| * Multiplication [[distributive law|distributes]] over addition:
| |
| :7. ''a'' ⋅ (''b'' + ''c'') = (''a'' ⋅ ''b'') + (''a'' ⋅ ''c'') for all ''a'', ''b'', ''c'' in ''R'' (left distributivity).
| |
| :8. (''b'' + ''c'') ⋅ ''a'' = (''b'' ⋅ ''a'') + (''c'' ⋅ ''a'') for all ''a'', ''b'', ''c'' in ''R'' (right distributivity).
| |
| | |
| ===Notes on the definition===
| |
| '''Warning:''' As explained in the [[Ring (mathematics)#History|history section]] below, many authors follow an alternative convention in which a ring is not required to have a 1. This article adopts the convention that, unless otherwise stated, a ring is assumed to have a 1. <!--- This is also the convention in [[Wikipedia:Manual of Style/Mathematics]]. ---> A structure satisfying all the axioms ''except'' the sixth (existence of a multiplicative identity 1) is called a '''[[rng (algebra)|rng]]''' (or sometimes '''[[pseudo-ring]]'''). For example, the set of even integers with the usual + and · is a rng, but not a ring.
| |
| | |
| The operations + and ⋅ are called ''addition'' and ''multiplication'', respectively. The multiplication symbol ⋅ is often omitted, so the mere [[wikt:juxtaposition|juxtaposition]] of ring elements is interpreted as multiplication. For example, ''xy'' means ''x''⋅''y''.
| |
| | |
| Although ring addition is [[commutative law|commutative]], ring multiplication is not required to be commutative: ''ab'' need not necessarily equal ''ba''. Rings that also satisfy commutativity for multiplication (such as the ring of [[integer]]s) are called '''[[commutative ring]]s'''. Books on commutative algebra or algebraic geometry often adopt the convention that "ring" means "commutative ring", to simplify terminology.
| |
| | |
| ===Basic properties===
| |
| Some basic properties of a ring follow immediately from the axioms:
| |
| * The additive identity, the additive inverse of each element, and the multiplicative identity are unique.
| |
| * For any element ''x'' in a ring ''R'', one has ''x''0 = 0 = 0''x'' and (–1)''x'' = –''x''.
| |
| * If 0 = 1 in a ring ''R'', then ''R'' has only one element, and is called the [[zero ring]].
| |
| * The [[binomial formula]] holds for any commuting pair of elements (i.e., any ''x'' and ''y'' such that ''xy'' = ''yx'').
| |
| | |
| ===Example: Integers modulo 4===
| |
| {{see also| Modular arithmetic}}
| |
| | |
| Equip the set <math>\mathbf{Z}_4 = \{\overline{0},\overline{1},\overline{2},\overline{3}\}</math> with the following operations:
| |
| | |
| * The sum <math>\overline{x} + \overline{y}</math> in '''Z'''<sub>4</sub> is the remainder when the integer ''x'' + ''y'' is divided by 4. For example, <math>\overline{2} + \overline{3} = \overline{1}</math> and <math>\overline{3} + \overline{3} = \overline{2}</math>.
| |
| | |
| * The product <math>\overline{x} \cdot \overline{y}</math> in '''Z'''<sub>4</sub> is the remainder when the integer ''xy'' is divided by 4. For example, <math>\overline{2} \cdot \overline{3} = \overline{2}</math> and <math>\overline{3} \cdot \overline{3} = \overline{1}</math>.
| |
| | |
| Then '''Z'''<sub>4</sub> is a ring: each axiom follows from the corresponding axiom for '''Z'''. If ''x'' is an integer, the remainder of ''x'' when divided by 4 is an element of '''Z'''<sub>4</sub>, and this element is often denoted by {{nowrap|"''x'' mod 4"}} or <math>\overline{x}</math>, which is consistent with the notation for 0,1,2,3. The additive inverse of any <math>\overline{x}</math> in '''Z'''<sub>4</sub> is <math>\overline{-x}</math>. For example, <math>-\overline{3}= \overline{-3} = \overline{1}.</math>
| |
| | |
| ===Example: 2-by-2 matrices===
| |
| {{main|Matrix ring}}
| |
| The set of 2-by-2 [[Matrix (mathematics)|matrices]] with [[real number]] entries is written
| |
| | |
| :<math>\mathcal{M}_2(\mathbb{R}) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \bigg|\ a,b,c,d \in \mathbb{R} \right\}. </math>
| |
| | |
| With the operations of matrix addition and [[matrix multiplication]], this set satisfies the above ring axioms. The element <math> \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}</math> is the multiplicative identity of the ring. If <math>A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}</math> and <math>B=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}</math>, then <math>AB=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}</math> while <math>BA=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}</math>; this example shows that the ring is noncommutative.
| |
| | |
| More generally, for any ring ''R'', commutative or not, and any nonnegative integer ''n'', one may form the ring of ''n''-by-''n'' matrices with entries in ''R'': see [[matrix ring]].
| |
| | |
| == History ==
| |
| | |
| {{See also|Ring theory#History}}
| |
| [[File:Dedekind.jpeg|thumb|100px|right|[[Richard Dedekind]], one of the founders of [[ring theory]].]]
| |
| | |
| ===Dedekind===
| |
| The study of rings originated from the theory of [[polynomial ring]]s and the theory of [[algebraic integer]]s.<ref name="history">[http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Ring_theory.html The development of Ring Theory]</ref> In 1871 [[Richard Dedekind]] defined the concept of the ring of integers of a number field.<ref>Kleiner 1998, p. 27.</ref> In this context, he introduced the terms "ideal" (inspired by [[Ernst Kummer]]'s notion of ideal number) and "module" and studied their properties. But Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.
| |
| | |
| ===Hilbert===
| |
| The term "Zahlring" (number ring) was coined by [[David Hilbert]] in 1892 and published in 1897.<ref>Hilbert 1897.</ref> In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (e.g., spy ring),<ref>[http://mathoverflow.net/questions/117292/why-is-a-ring-called-a-ring]</ref> so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself.<ref>{{Citation|last=Cohn|first=Harvey|title=Advanced Number Theory|publisher=Dover Publications|location=New York|year=1980|page=49|isbn=978-0-486-64023-5}}</ref> Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if ''a''<sup>3</sup> − 4''a'' + 1 = 0 then ''a''<sup>3</sup> = 4''a'' − 1, ''a''<sup>4</sup> = 4''a''<sup>2</sup> − ''a'', ''a''<sup>5</sup> = −''a''<sup>2</sup> + 16''a'' − 4, ''a''<sup>6</sup> = 16''a''<sup>2</sup> − 8''a'' + 1, ''a''<sup>7</sup> = −8''a''<sup>2</sup> + 65''a'' − 16, and so on; in general, ''a''<sup>''n''</sup> is going to be an integral linear combination of 1, ''a'', and ''a''<sup>2</sup>.
| |
| | |
| ===Fraenkel and Noether===
| |
| The first axiomatic definition of a ring was given by [[Adolf Fraenkel]] in 1914,<ref>Fraenkel, pp. 143–145</ref><ref>Jacobson (2009), p. 86, footnote 1.</ref> but his axioms were stricter than those in the modern definition. For instance, he required every [[zero divisor|non-zero-divisor]] to have a [[multiplicative inverse]].<ref>Fraenkel, p. 144, axiom ''R''<sub>8)</sub>.</ref> In 1921, [[Emmy Noether]] gave the modern axiomatic definition of [[commutative ring|(commutative) ring]] and developed the foundations of commutative ring theory in her monumental paper ''Idealtheorie in Ringbereichen''.<ref name="Noether, p. 29">Noether, p. 29.</ref>
| |
| | |
| ===Multiplicative identity: mandatory or optional?===
| |
| Fraenkel required a ring to have a multiplicative identity 1,<ref>Fraenkel, p. 144, axiom ''R''<sub>7)</sub>.</ref> whereas Noether did not.<ref name="Noether, p. 29"/>
| |
| | |
| Most or all books on algebra<ref>Van der Waerden, 1930.</ref><ref>Zariski and Samuel, 1958.</ref> up to around 1960 followed Noether's convention of not requiring a 1. Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of ring, especially in advanced books by notable authors such as Artin,<ref>Artin, p. 346.</ref> Atiyah and MacDonald,<ref>Atiyah and MacDonald, p. 1.</ref> Bourbaki,<ref>Bourbaki, p. 96.</ref> Eisenbud,<ref>Eisenbud, p. 11.</ref> and Lang.<ref>Lang, p. 83.</ref> But even today, there remain many books that do not require a 1.
| |
| | |
| Faced with this terminological ambiguity, some authors have tried to impose their views, while others have tried to adopt more precise terms.
| |
| | |
| In the first category, we find for instance Gardner and Wiegandt, who argue that if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and the fact that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable."<ref>Gardner and Wiegandt 2003.</ref>
| |
| | |
| In the second category, we find authors who use the following terms:<ref>Wilder 1965, p. 176.</ref><ref>Rotman 1998, p. 7.</ref>
| |
| | |
| :* rings with multiplicative identity: ''unital ring'', ''unitary ring'', ''ring with unity'', ''ring with identity'', or ''ring with 1''
| |
| :* rings without multiplicative identity: ''[[rng (algebra)|rng]]'' or ''[[pseudo-ring]]''.
| |
| | |
| == Basic examples ==
| |
| | |
| '''Commutative rings''':
| |
| * The motivating example is the ring of [[integer]]s with the two operations of addition and multiplication.
| |
| * The rational, real and complex numbers form commutative rings (in fact, they are even [[field (mathematics)|field]]s).
| |
| * The [[Gaussian integers]] form a ring, as do the [[Eisenstein integers]]. So does their generalization [[Kummer ring]]. cf. [[quadratic integers]].
| |
| * The set of all [[algebraic integer]]s forms a ring. This follows for example from the fact that it is the [[integral closure]] of the ring of rational integers in the field of complex numbers. The rings in the previous example are subrings of this ring.
| |
| * The [[polynomial ring]] ''R''[''X''] of polynomials over a ring ''R'' is also a ring.
| |
| * The set of [[formal power series]] ''R''<nowiki>[[</nowiki>''X''<sub>1</sub>, …, ''X''<sub>''n''</sub><nowiki>]]</nowiki> over a commutative ring ''R'' is a ring.
| |
| * If ''S'' is a set, then the [[power set]] of ''S'' becomes a ring if we define addition to be the [[symmetric difference]] of sets and multiplication to be [[intersection (set theory)|intersection]]. This corresponds to a [[ring of sets]] and is an example of a [[Boolean ring]].
| |
| * The set of all [[continuous function|continuous]] real-valued [[function (mathematics)|function]]s defined on the real line forms a commutative ring. The operations are addition and multiplication of functions.
| |
| | |
| '''Noncommutative rings''':
| |
| * For any ring ''R'' and any natural number ''n'', the set of all square ''n''-by-''n'' [[matrix (mathematics)|matrices]] with entries from ''R'', forms a ring with matrix addition and matrix multiplication as operations. For ''n'' = 1, this matrix ring is isomorphic to ''R'' itself. For ''n'' > 1 (and ''R'' not the zero ring), this matrix ring is noncommutative.
| |
| * If ''G'' is an [[abelian group]], then the [[group homomorphism|endomorphisms]] of ''G'' form a ring, the [[endomorphism ring]] End(''G'') of ''G''. The operations in this ring are addition and composition of endomorphisms.
| |
| * If ''G'' is a [[group (mathematics)|group]] and ''R'' is a ring, the [[group ring]] of ''G'' over ''R'' is a [[free module]] over ''R'' having ''G'' as basis. Multiplication is defined by the rules that the elements of ''G'' commute with the elements of ''R'' and multiply together as they do in the group ''G''.
| |
| * Many rings that appear in analysis are noncommutative. For example, most [[Banach algebra]]s are noncommutative.
| |
| | |
| '''Non-examples:'''
| |
| *The set of [[natural number]]s '''N''' with the usual operations is not a ring, since ('''N''', +) is not even a [[group (mathematics)|group]] (the elements are not all [[invertible]] with respect to [[addition]]). For instance, there is no [[natural number]] which can be added to 3 to get 0 as a result. There is a natural way to make it a ring by adding negative numbers to the set, thus obtaining the ring of [[integer]]s. The natural numbers (including 0) form an algebraic structure known as a [[semiring]] (which has all of the properties of a ring except the additive inverse property).
| |
| * Let ''R'' be the set of all continuous functions on the real line that vanish outside a bounded interval depending on the function, with addition as usual but with multiplication defined as [[convolution]]:
| |
| *:<math>(f * g)(x) = \int_{-\infty}^{\infty} f(y)g(x-y)dy.</math>
| |
| :Then ''R'' is a [[rng (algebra)|rng]], but not a ring: the [[Dirac delta function]] has the property of a multiplicative identity, but it is not a function and hence is not an element of ''R''.
| |
| | |
| == Basic concepts ==
| |
| | |
| ===Elements in a ring===
| |
| A left [[zero divisor]] is an element <math>a</math> of a ring <math>R</math> such that there exists a nonzero element <math>b</math> of <math>R</math> such that <math>ab = 0</math>.<ref>This is the definition of Bourbaki. Some other authors such as Lang require a zero divisor to be nonzero.</ref> A right zero divisor is defined similarly.
| |
| | |
| A [[nilpotent element]] is an element <math>a</math> such that <math>a^n = 0</math> for some <math>n > 0</math>. One example of a nilpotent element is a [[nilpotent matrix]]. A nilpotent element in a [[zero ring|nonzero ring]] is necessarily a zero divisor.
| |
| | |
| An [[idempotent element|idempotent]] <math>e</math> is an element such that <math>e^2 = e</math>. One example of an idempotent element is a [[projection (linear algebra)|projection]] in linear algebra.
| |
| | |
| A [[unit (ring theory)|unit]] is an element <math>a</math> having a [[multiplicative inverse]]; in this case the inverse is unique, and is denoted by <math>a^{-1}</math>. The set of units of a ring is a [[group (mathematics)|group]] under ring multiplication; this group is denoted by <math>R^\times</math> or <math>R^*</math> or <math>U(R)</math>. For example, if ''R'' is the ring of all square matrices of size ''n'' over a field, then <math>R^\times</math> consists of the set of all invertible matrices of size ''n'', and is called the [[general linear group]].
| |
| | |
| ===Subring===
| |
| {{main|Subring}}
| |
| A subset ''S'' of ''R'' is said to be a [[subring]] if it can be regarded as a ring with the addition and the multiplication [[restricted function|restricted]] from ''R'' to ''S''. Equivalently, ''S'' is a subring if it is not empty, and for any ''x'', ''y'' in ''S'', <math>xy</math>, <math>x+y</math> and <math>-x</math> are in ''S''. If all rings have been assumed, by convention, to have a multiplicative identity, then to be a subring one would also require ''S'' to share the same identity element as ''R''.<ref>In the unital case, like addition and multiplication, the multiplicative identity must be restricted from the original ring. The definition is also equivalent to requiring the set-theoretic inclusion is a ring homomorphism.</ref> So if all rings have been assumed to have a multiplicative identity, then a proper ideal is not a subring.
| |
| | |
| For example, the ring '''Z''' of [[integer]]s is a subring of the [[field (mathematics)|field]] of [[real number]]s and also a subring of the ring of [[polynomial]]s '''Z'''[''X''] (in both cases, '''Z''' contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers 2'''Z''' does not contain the identity element 1 and thus does not qualify as a subring.
| |
| | |
| The intersection of subrings is a subring. The smallest subring containing a given subset ''E'' of ''R'' is called a subring generated by ''E''. Such a subring exists since it is the intersection of all subrings containing ''E''.
| |
| | |
| For a ring ''R'', the smallest subring containing 1 is called the ''characteristic subring'' of ''R''. It can be obtained by adding copies of 1 and −1 together many times in any mixture. It is possible that <math>n\cdot 1=1+1+\ldots+1</math> (''n'' times) can be zero. If ''n'' is the smallest positive integer such that this occurs, then ''n'' is called the ''[[Characteristic (algebra)|characteristic]]'' of ''R''. In some rings, <math>n\cdot 1</math> is never zero for any positive integer ''n'', and those rings are said to have ''characteristic zero''.<!-- By using a homomorphism from the integers into ''R'' that sends 1 to 1, it can be shown by an isomorphism theorem that the characteristic subring is always a quotient ring of the integers. The problem: we haven't introduced homomorphisms and quotient rings yet. -->
| |
| | |
| Given a ring ''R'', let <math>\operatorname{Z}(R)</math> denote the set of all elements ''x'' in ''R'' such that ''x'' commutes with every element in ''R'': <math>xy = yx</math> for any ''y'' in ''R''. Then <math>\operatorname{Z}(R)</math> is a subring of ''R''; called the [[center (algebra)|center]] of ''R''. More generally, given a subset ''X'' of ''R'', let ''S'' be the set of all elements in ''R'' that commute with every element in ''X''. Then ''S'' is a subring of ''R'', called the [[centralizer (ring theory)|centralizer]] (or commutant) of ''X''. The center is the centralizer of the entire ring ''R''. Elements or subsets of the center are said to be ''central'' in ''R''; they generate a subring of the center.
| |
| | |
| ===Ideal===
| |
| {{main|Ideal (ring theory)}}
| |
| The definition of an [[ideal (ring theory)|ideal]] in a ring is analogous to that of [[normal subgroup]] in a group. But, in actuality, it plays a role of an idealized generalization of an element in a ring; hence, the name "ideal". Like elements of rings, the study of ideals is central to structural understanding of a ring.
| |
| | |
| Let ''R'' be a ring. A subset ''I'' of ''R'' is then said to be a '''left ideal''' in ''R'' if <math>R I \subseteq I</math>. Here, <math>R I</math> denotes the span of ''I'' over ''R''; i.e., the set of finite sums
| |
| :<math>r_1 x_1 + \cdots + r_n x_n, \quad r_i \in R, \quad x_i \in I.</math>
| |
| Similarly, ''I'' is said to be '''right ideal''' if <math>I R \subseteq I</math>. A subset ''I'' is said to be a '''two-sided ideal''' or simply '''ideal''' if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of ''R''. If ''E'' is a subset of ''R'', then <math>R E</math> is a left ideal, called the left ideal generated by ''E''; it is the smallest left ideal containing ''E''. Similarly, the two-sided ideal generated by ''E'' is the smallest two-sided ideal containing ''E'', or, equivalently, <math>R E R.</math>
| |
| | |
| Given right (or left, or two-sided) ideals ''A'' and ''B'' of ''R'', it is possible to show that the [[set intersection]] of ''A'' with ''B'' is an ideal of the same type as ''A'' and ''B''. It is also possible to define a [[product of ideals]] such that ''AB'' is another ideal of the same sidedness as ''A'' and ''B''. The [[sum of ideals]] ''A''+''B'' is also an ideal of the same type as ''A'' and ''B''.
| |
| | |
| If ''x'' is in ''R'', then <math>Rx</math> and <math>xR</math> are left ideals and right ideals, respectively; they are called the [[principal ideal|principal]] left ideals and right ideals generated by ''x''. The principal ideal <math>RxR</math> is written as <math>(x)</math>. For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal.
| |
| | |
| Like a group, a ring is said to be a [[simple ring|simple]] if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.
| |
| | |
| Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite [[total order#Chains|chain]] of left ideals is called a left [[Noetherian ring]]. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left [[Artinian ring]]. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the [[Hopkins–Levitzki theorem]]). The integers, however, form a Noetherian ring which is not Artinian.
| |
| | |
| For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal ''P'' of ''R'' is called a [[prime ideal]] if for any elements <math>x, y\in R</math> we have that <math>xy \in P</math> implies either <math>x \in P</math> or <math>y\in P</math>. Equivalently, ''P'' is prime if for any ideals <math>I, J</math> we have that <math>IJ \subseteq P</math> implies either <math>I \subseteq P</math> or <math>J \subseteq P.</math> This later formulation illustrates the idea of ideals as generalizations of elements.
| |
| | |
| ===Homomorphism===
| |
| {{main|Ring homomorphism}}
| |
| A '''[[ring homomorphism|homomorphism]]''' from a ring (''R'', +, '''·''') to a ring (''S'', ‡, *) is a function ''f'' from ''R'' to ''S'' that preserves the ring operations; namely, such that, for all ''a'', ''b'' in ''R'' the following identities hold:
| |
| | |
| * ''f''(''a'' + ''b'') = ''f''(''a'') ‡ ''f''(''b'')
| |
| | |
| * ''f''(''a'' '''·''' ''b'') = ''f''(''a'') * ''f''(''b'')
| |
| | |
| * ''f''(1<sub><sub>''R''</sub></sub>) = 1<sub><sub>''S''</sub></sub>
| |
| | |
| If one is working with not necessarily unital rings, then the third condition is dropped.
| |
| | |
| A ring homomorphism is said to be an '''isomorphism''' if there exists an inverse homomorphism to ''f'' (i.e., a ring homomorphism which is an [[inverse function]]). Any [[bijection|bijective]] ring homomorphism is a ring isomorphism. Two rings <math>R, S</math> are said to be isomorphic if there is an isomorphism between them and in that case one writes <math>R \simeq S</math>. A ring homomorphism between the same ring is called an endomorphism and an isomorphism between the same ring an automorphism.
| |
| | |
| Examples:
| |
| *The function that maps each integer ''x'' to its remainder modulo 4 (a number in {0, 1, 2, 3}) is a homomorphism from the ring '''Z''' to the quotient ring '''Z'''/4'''Z''' ("quotient ring" is defined below).
| |
| *If <math>u</math> is a unit element in a ring ''R'', then <math>R \to R, x \mapsto uxu^{-1}</math> is a ring homomorphism, called an [[inner automorphism]] of ''R''.
| |
| *Let ''R'' be a commutative ring of prime characteristic ''p''. Then <math>x \mapsto x^p</math> is a ring endmorphism of ''R'' called the [[Frobenius homomorphism]].
| |
| *The [[Galois group]] of a field extension <math>L/K</math> is the set of all automorphisms of ''L'' whose restrictions to ''K'' are the identity.
| |
| * For any ring ''R'', there are a unique ring homomorphism '''Z''' →''R'' and a unique ring homomorphism ''R'' →0.
| |
| | |
| Given a ring homomorphism <math>f:R \to S</math>, the set of all elements mapped to 0 by ''f'' is called the [[kernel of a ring homomorphism|kernel]] of ''f''. It is a two-sided ideal of ''R''. The image of ''f'', on the other hand, is not always an ideal, but it is always a subring of ''S''.
| |
| | |
| To give a ring homomorphism from a commutative ring ''R'' to a ring ''A'' with image contained in the center of ''A'' is the same as to give a structure of an [[associative algebra|algebra]] over ''R'' to ''A'' (in particular gives a structure of ''A''-module). Indeed, if ''f'' is such a map, one can define the scalar multiplication by ''R'' on ''A'' as <math>f(r)a</math>. Conversely, if ''A'' is an algebra over ''R'', then <math>R \to A, r \mapsto r 1</math> is such a ring homomorphism.
| |
| | |
| ===Quotient ring===
| |
| {{main|Quotient ring}}
| |
| The '''[[quotient ring]]''' of a ring, is analogous to the notion of a [[quotient group]] of a group. More formally, given a ring (''R'', +, '''·''' ) and a [[Ideal (ring theory)|two-sided ideal]] ''I'' of (''R'', +, '''·''' ), the '''quotient ring''' (or '''factor ring''') ''R/I'' is the set of cosets of ''I'' (with respect to the [[additive group]] of (''R'', +, '''·''' ); i.e. cosets with respect to (''R'', +)) together with the operations:
| |
| | |
| :(''a'' + ''I'') + (''b'' + ''I'') = (''a'' + ''b'') + ''I'' and
| |
| :(''a'' + ''I'')(''b'' + ''I'') = (''ab'') + ''I''.
| |
| | |
| for every ''a'', ''b'' in ''R''.
| |
| | |
| Like the case of a quotient group, there is a canonical map <math>p: R \to R/I</math> given by <math>x \mapsto x + I</math>. It is surjective and satisfies the universal property: if <math>f:R \to S</math> is a ring homomorphism such that <math>f(I) = 0</math>, then there is a unique <math>\overline{f}: R/I \to S</math>
| |
| such that <math>f = \overline{f} \circ p</math>. In particular, taking ''I'' to be the kernel, one sees that the quotient ring <math>R / \operatorname{ker} f</math> is isomorphic to the image of ''f''; the fact known as the first [[isomorphism theorem]]. The last fact implies that actually ''any'' surjective ring homomorphism satisfies the universal property since the image of such a map is a quotient ring.
| |
| | |
| A subset of ''R'' and the quotient <math>R/I</math> are related in the following way. A subset of ''R'' is called a [[system of representatives]] of <math>R/I</math> if no two elements in the set belong to the same coset; i.e., each element in the set represents a unique coset. It is said to be complete if the restriction of <math>R \to R/I</math> to it is surjective.
| |
| | |
| == Constructions ==
| |
| | |
| === Direct product ===
| |
| {{Main|Direct product of rings}}
| |
| | |
| Let ''R'' and ''S'' be rings. Then the [[cartesian product|product]] {{nowrap|''R'' × ''S''}} can be equipped with the following natural ring structure:
| |
| | |
| * (''r''<sub>1</sub>, ''s''<sub>1</sub>) + (''r''<sub>2</sub>, ''s''<sub>2</sub>) = (''r''<sub>1</sub> + ''r''<sub>2</sub>, ''s''<sub>1</sub> + ''s''<sub>2</sub>)
| |
| | |
| * (''r''<sub>1</sub>, ''s''<sub>1</sub>) ⋅ (''r''<sub>2</sub>, ''s''<sub>2</sub>) = (''r''<sub>1</sub> ⋅ ''r''<sub>2</sub>, ''s''<sub>1</sub> ⋅ ''s''<sub>2</sub>)
| |
| | |
| for every ''r''<sub>1</sub>, ''r''<sub>2</sub> in ''R'' and ''s''<sub>1</sub>, ''s''<sub>2</sub> in ''S''. The ring {{nowrap|''R'' × ''S''}} with the above operations of addition and multiplication and the multiplicative identity <math>(1, 1)</math> is called the '''[[Direct product of rings|direct product]]''' of ''R'' with ''S''. The same construction also works for an arbitrary family of rings: if <math>R_i</math> are rings indexed by a set ''I'', then <math>\prod_{i \in I} R_i</math> is a ring with componentwise addition and multiplication.
| |
| | |
| A "finite" direct product may also be viewed as a direct sum of ideals.<ref>{{harvnb|Cohn|2003|loc=Theorem 4.5.1}}</ref> Namey, let <math>R_i, 1 \le i \le n</math> be rings, <math>R_i \to R = \prod R_i</math> the inclusions with the images <math>\mathfrak{a}_i</math> (in particular <math>\mathfrak{a}_i</math> are rings though not subrings). Then <math>\mathfrak{a}_i</math> are ideals of ''R'' and
| |
| :<math>R = \mathfrak{a}_1 \oplus \cdots \oplus \mathfrak{a}_n, \quad \mathfrak{a}_i \mathfrak{a}_j = 0, i \ne j, \quad \mathfrak{a}_i^2 \subseteq \mathfrak{a}_i</math>
| |
| as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to ''R''. Equivalently, the above can be done through [[central idempotent]]s. Assume ''R'' has the above decomposition. Then we can write
| |
| :<math>1 = e_1 + \cdots + e_n, \quad e_i \in \mathfrak{a}_i.</math>
| |
| By the conditions on <math>\mathfrak{a}_i</math>, one has that <math>e_i</math> are central idempotents and <math>e_i e_j = 0, i \ne j</math> (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let <math>\mathfrak{a}_i = R e_i</math>, which are two-sided ideals. If each <math>e_i</math> is not a sum of orthogonal central idempotents,<ref>such a central idempotent is called [[centrally primitive]].</ref> then their direct sum is isomorphic to ''R''.
| |
| | |
| Let ''R'' be a commutative ring and <math>\scriptstyle \mathfrak{a}_1, \cdots, \mathfrak{a}_n</math> be ideals such that <math>\mathfrak{a}_i + \mathfrak{a}_j = (1)</math> whenever <math>i \ne j</math>. Then the [[Chinese remainder theorem]] says that
| |
| :<math>R / \cap \mathfrak{a}_i \simeq \prod R/ \mathfrak{a}_i, \quad x \mapsto (x \text{ mod } \mathfrak{a}_1, \ldots , x \text{ mod } \mathfrak{a}_n)</math>.
| |
| | |
| An important application of an infinite direct product is the construction of a [[projective limit]] of rings, which carries over in verbatim from that for groups. Namely, suppose we're given a family of rings <math>R_i</math>, ''i'' running over positive integers, say, and ring homomorphisms <math>R_j \to R_i, j \ge i</math> such that <math>R_i \to R_i</math> are all the identities and <math>R_k \to R_j \to R_i</math> is <math>R_k \to R_i</math> whenever <math>k \ge j \ge i</math>. Then <math>\varprojlim R_i</math> is the subring of <math>\prod R_i</math> consisting of <math>(x_n)</math> such that <math>x_j</math> maps to <math>x_i</math> under <math>R_j \to R_i, j \ge i</math>.
| |
| | |
| The notion of a '''[[subdirect product]]''' of rings generalizes a direct product of rings.
| |
| | |
| ===Polynomial ring===
| |
| {{main|Polynomial ring}}
| |
| | |
| Given a symbol ''t'' (called a variable) and a commutative ring ''R'', the set of polynomials
| |
| | |
| : <math>R[t] = \left\{ a_n t^n + a_{n-1} t^{n -1} + \dots + a_1 t + a_0 \mid n \ge 0, a_j \in R \right\}</math>
| |
| | |
| forms a commutative ring with the usual addition and multiplication, containing ''R'' as a subring. It is called the [[polynomial ring]] over ''R''. More generally, the set <math>R[t_1, \ldots, t_n]</math> of all polynomials in variables <math>t_1, \ldots, t_n</math> forms a commutative ring, containing <math>R[t_i]</math> as subrings.
| |
| | |
| If ''R'' is an integral domain, then <math>R[t]</math> is also an integral domain; its field of fractions is the field of [[rational function]]s. If ''R'' is a noetherian ring, then <math>R[t]</math> is a noetherian ring. If ''R'' is a unique factorization domain, then <math>R[t]</math> is a unique factorization domain. Finally, ''R'' is a field if and only if <math>R[t]</math> is a principal ideal domain.
| |
| | |
| Let <math>R \subseteq S</math> be commutative rings. Given an element ''x'' of ''S'', one can consider the ring homomorphism
| |
| :<math>R[t] \to S, \quad f \mapsto f(x)</math>
| |
| (i.e., the [[Substitution (algebra)|substitution]]). If ''S''=''R''[''t''] and ''x''=''t'', then ''f''(''t'')=''f''. Because of this, the polynomial ''f'' is often also denoted by <math>f(t)</math>. The image of the map <math>f \mapsto f(x)</math> is denoted by <math>R[x]</math>; it is the same thing as the subring of ''S'' generated by ''R'' and ''x''.
| |
| | |
| The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism <math>\phi: R \to S</math> and an element ''x'' in ''S'' there exists a unique ring homomorphism <math>\overline{\phi}: R[t] \to S</math> such that <math>\overline{\phi}(t) = x</math> and <math>\overline{\phi}</math> restricts to <math>\phi</math>.<ref>{{harvnb|Jacobson|1974|loc=Theorem 2.10}}</ref> If a monic polynomial generates the kernel of <math>\overline{\phi}</math>, it is called the [[minimal polynomial (field theory)|minimal polynomial]] of ''x'' over ''R''. In a module-theoretic language, the universal property says that <math>R[x]</math> is a free module over ''R'' with generators <math>1, x, x^2, \dots</math>.
| |
| | |
| To give an example, let ''S'' be the ring of all functions from ''R'' to itself; the addition and the multiplication are those of functions. Let ''x'' be the identity function. Each ''r'' in ''R'' defines a constant function, giving rise to the homomorphism <math>R \to S</math>. The universal property says that this map extends uniquely to
| |
| :<math>R[t] \to S, \quad f \mapsto \overline{f}</math>
| |
| (''t'' maps to ''x'') where <math>\overline{f}</math> is the [[polynomial function]] defined by ''f''. The resulting map is injective if and only if ''R'' is infinite.
| |
| | |
| Given a non-constant monic polynomial ''f'' in <math>R[t]</math>, there exists a ring ''S'' containing ''R'' such that ''f'' is a product of linear factors in <math>S[t]</math>.<ref>{{harvnb|Bourbaki|Algèbre commutative|loc=Ch 5. §1, Lemma 2}}</ref>
| |
| | |
| Let ''k'' be an algebraically closed field. The [[Hilbert's Nullstellensatz]] (theorem of zeros) states that there is a one-to-one correspondence between the set of all prime ideals in <math>k[t_1, \ldots, t_n]</math> and the set of closed subvarieties of <math>k^n</math>. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf. [[Gröbner basis]].)
| |
| | |
| A quotient ring of the polynomial ring <math>R[t_1, \cdots, t_n]</math> is said to be a [[finitely generated algebra]] over ''R'' or of finite type over ''R''. The [[Noether normalization lemma]] says that any finitely generated commutative ''k''-algebra ''R'' contains the polynomial ring with the coefficients in ''k'' over which ''R'' is finitely generated as a module. A polynomial ring is relatively well-understood and thus the theorem allows one to study a ring from the known facts about a polynomial ring.
| |
| | |
| There are some other related constructions. A [[formal power series ring]] <math>R[\![t]\!]</math> consists of formal power series
| |
| :<math>\sum_0^\infty a_i t^i, \quad a_i \in R</math>
| |
| together with multiplication and addition that mimic those for convergent series. It contains <math>R[t]</math> as a subring. The important advantage of a formal power series ring is that it is [[local ring|local]] (in fact, [[complete ring|complete]]). One can also consider a polynomial ring in infinitely many variables <math>R[t_1, t_2, \dots]</math>: it is a union (i.e., direct limit) of <math>R[t_1, t_2, \dots, t_n]</math> over all ''n''. This ring is often used to furnish counterexamples.
| |
| | |
| Finally, there is a closely related notion: [[ring of polynomial functions]] on a vector space ''V''. If ''V'' is a vector space over an infinite field, then, by choosing a basis, it may be identified with a polynomial ring.
| |
| | |
| ===Matrix ring and endomorphism ring ===
| |
| {{Main|Matrix ring|Endomorphism ring}}
| |
| | |
| Let ''R'' be a ring (not necessarily commutative). The set of all square matrices of size ''n'' with entries in ''R'' forms a ring with the entry-wise addition and the usual [[matrix multiplication]]. It is called the [[matrix ring]] and is denoted by M<sub>''n''</sub>(''R''). Given a right ''R''-module <math>U</math>, the set of all ''R''-linear maps from ''U'' to itself forms a ring with addition that is of function and multiplication that is of [[composition of functions]]; it is called the endomorphism ring of ''U'' and is denoted by <math>\operatorname{End}_R(U)</math>. As in linear algebra, a matrix ring may be interpreted as an endomorphism ring: <math>\operatorname{End}_R(R^n) \simeq \operatorname{M}_n(R)</math>. This is a special case of the following fact: If <math>f: \oplus_1^n U \to \oplus_1^n U</math> is an ''R''-linear map, then ''f'' may be written as a matrix with entries <math>f_{ij}</math> in <math>S = \operatorname{End}_R(U)</math>, resulting in a homomorphism
| |
| :<math>\operatorname{End}_R(\oplus_1^n U) \to \operatorname{M}_n(S), \quad f \mapsto (f_{ij})</math>
| |
| which is clearly an isomorphism.
| |
| | |
| Let <math>e_{ij}</math> be a matrix whose <math>(i, j)</math>-th entry is 1 and the other entries zero. If ''C'' is the centralizer in ''R'' of <math>e_{ij}</math>'s, then <math>R \simeq \operatorname{M}_n(C)</math>.<ref name="Cohn matrix">{{harvnb|Cohn|2003|loc=4.4}}</ref>
| |
| | |
| Any ring homomorhism ''R'' → ''S'' induces {{nowrap|M<sub>''n''</sub>(''R'') → M<sub>''n''</sub>(''S'')}}; in fact, any ring homomorphism between matrix rings arises in this way.<ref name="Cohn matrix" />
| |
| | |
| [[Schur's lemma]] says that if ''U'' is a simple right ''R''-module, then <math>\operatorname{End}_R(U)</math> is a division ring.<ref>{{harvnb|Lang|2002|loc=Ch. XVII. Proposition 1.1.}}</ref> Let <math>\displaystyle U = \bigoplus_{i = 1}^r m_i U_i</math> be a direct sum of ''R''-modules where <math>U_i</math> are simple modules and <math>mU_i</math> means a direct sum of ''m'' copies of <math>U_i</math>. Then
| |
| :<math>\operatorname{End}_R(U) \simeq \bigoplus_1^r \operatorname{M}_{m_i} (\operatorname{End}_R(U_i))</math>.
| |
| The [[Artin–Wedderburn theorem]] states any [[semisimple ring]] (cf. below) is of this form.
| |
| | |
| A ring ''R'' and the matrix ring M<sub>''n''</sub>(''R'') over it are [[Morita equivalent]]: the category of right modules of ''R'' is equivalent to the category of right modules over M<sub>''n''</sub>(''R'').<ref name="Cohn matrix" /> In particular, two-sided ideals in ''R'' correspond in one-to-one to two-sided ideals in M<sub>''n''</sub>(''R'').
| |
| | |
| Examples:
| |
| *The automorphisms of the [[projective line over a ring]] are given by [[homography|homographies]] from the 2 x 2 matrix ring.
| |
| | |
| The following construction is useful for an application to linear algebra (see also another example in the "Domain" section below.) Let ''V'' be a left ''R''-module, <math>\phi: R \to S = \operatorname{End}_R(V)</math> be given by <math>\phi(r)v = rv</math> and <math>T: V \to V</math> a linear map. Then <math>\phi(R)</math> and ''T'' generate a commutative subring of ''S''. By the universal property of a polynomial ring, <math>\phi</math> uniquely extends to
| |
| :<math>R[t] \to S, \quad f \mapsto f(T)</math>.
| |
| In particular, ''V'' acquires a structure of a module over <math>R[t]</math>; let <math>V_T</math> denote the resulting module. This allows one to study ''T'' in terms of the module. For example, assuming ''R'' is a field, ''T'' is diagonalizable as a linear transformation if and only if <math>V_T</math> is a semisimple module. For another example, <math>V_T, V_U</math> are isomorphic as a module if and only if <math>T, U</math> are similar; i.e., <math>T = H \circ U \circ H^{-1}</math> for some isomorphism ''H''.
| |
| <!--Let ''R'' be a ring (with unity) and ''V'' a left ''R''-module. Put <math>R' = \operatorname{End}_R(V)</math>. We can view ''V'' as a left ''R'''-module through the action <math>\phi \cdot v = \phi(v).</math> <math>R'</math> is called the '''commutant''' of ''R''. Let <math>R''</math> be the double commutant of ''R''; i.e., <math>R'' = \operatorname{End}_{R'}(V)</math>. The density theorem<ref>{{harvnb|Lang|2002|loc=Ch XVII, § 3}}</ref> then says the image of the canonical map <math>R \to R'', \, a \mapsto l_a; l_a(v) = av</math> is "large" in the sense as follows. Assume ''V'' is semisimple. Then given a ''f'' in ''R'' and vectors <math>v_1, \dots, v_n</math> in ''V'', there exists an element ''a'' in ''R'' such that
| |
| :<math>a v_i = f(v_i), \quad 1 \le i \le n.</math> -->
| |
| | |
| === Localization ===
| |
| The [[localization of a ring|localization]] generalizes the construction of the [[field of fractions]] of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring ''R'', the localization of ''R'' with respect to a subset ''S'' of ''R'' is a ring <math>R[S^{-1}]</math> together with the ring homomorphism <math>R \to R[S^{-1}]</math> that "inverts" ''S''; that is, the homomorphism sends elements in ''S'' to the unit elements in <math>R[S^{-1}]</math>, and, moreover, any ring homomorphism from ''R'' that "inverts" ''S'' uniquely factors through <math>R[S^{-1}]</math>. For example, if ''R'' is a commutative ring and ''f'' an element in ''R'', then the localization <math>R[f^{-1}]</math> consists of elements of the form <math>r/f^n, \, r \in R , \, n \ge 0</math> (to be precise, <math>R[f^{-1}] = R[t]/(tf - 1).</math>)<ref>{{harvnb|Eisenbud|2004|loc=Exercise 2.2}}</ref>
| |
| | |
| The localization is frequently applied to a commutative ring ''R'' with respect to the complement of a prime ideal (or a union of prime ideals) in ''R''. In that case <math>S = R - \mathfrak{p}</math>, one often writes <math>R_\mathfrak{p}</math> for <math>R[S^{-1}]</math>. <math>R_\mathfrak{p}</math> is then a local ring with the maximal ideal <math>\mathfrak{p} R_\mathfrak{p}</math>. This is the reason for the terminology "localization". The field of fractions of an integral domain ''R'' is the localization of ''R'' at the prime ideal zero. If
| |
| <math>\mathfrak{p}</math> is a prime ideal of a commutative ring ''R'', then the field of fractions of <math>R/\mathfrak{p}</math> is the same as the residue field of the local ring <math>R_\mathfrak{p}</math> and is denoted by <math>k(\mathfrak{p})</math>.<!-- In algebraic geometry, the field of fractions is the localization at the "[[generic point]]" -->
| |
| | |
| If ''M'' is a left ''R''-module, then the localization of ''M'' with respect to ''S'' is given by a [[change of rings]] <math>M[S^{-1}] = R[S^{-1}] \otimes_R M</math>.<!-- intuitively, M_p corresponds to the fiber over ''p'' in ''M''. --> In particular, there is the canonical map <math>M \to M[S^{-1}], \, m \mapsto m / 1</math>. Its kernel consists of elements ''m'' such that <math>sm = 0</math> for some ''s'' in ''S''.
| |
| | |
| The most important properties of localization are the following: when ''R'' is a commutative ring and ''S'' multiplicatively closed subset
| |
| *<math>\mathfrak{p} \mapsto \mathfrak{p}[S^{-1}]</math> is a bijection between the set of all prime ideals in ''R'' disjoint from ''S'' and the set of all prime ideals in <math>R[S^{-1}]</math>.<ref>{{harvnb|Milne|2012|loc=Proposition 6.4}}</ref>
| |
| *<math>R[S^{-1}] = \varinjlim R[f^{-1}]</math>, ''f'' running over elements in ''S'' with partial ordering given by divisibility.<ref>{{harvnb|Milne|2012|loc=The end of Chapter 7}}</ref>
| |
| *The localization is exact:
| |
| *:<math>0 \to M'[S^{-1}] \to M[S^{-1}] \to M''[S^{-1}] \to 0</math> is exact over <math>R[S^{-1}]</math> whenever <math>0 \to M' \to M \to M'' \to 0</math> is exact over ''R''.
| |
| *Conversely, if <math>0 \to M'_\mathfrak{m} \to M_\mathfrak{m} \to M''_\mathfrak{m} \to 0</math> is exact for any maximal ideal <math>\mathfrak{m}</math>, then <math>0 \to M' \to M \to M'' \to 0</math> is exact.
| |
| *A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.)
| |
| | |
| In category theory, a [[localization of a category]] amounts to making some morphisms isomorphisms. An element in a commutative ring ''R'' may be thought of as an endomorphism of any ''R''-module. Thus, categorically, a localization of ''R'' with respect to a subset ''S'' of ''R'' is a functor from the category of ''R''-modules to itself that sends elements of ''S'' viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, ''R'' then maps to <math>R[S^{-1}]</math> and ''R''-modules map to <math>R[S^{-1}]</math>-modules.)
| |
| | |
| === Completion ===
| |
| Let ''R'' be a commutative ring, and let ''I'' be an ideal of ''R''.
| |
| The '''[[Completion (ring theory)|completion]]''' of ''R'' at ''I'' is the projective limit <math>\hat{R} = \varprojlim R/I^n</math>; it is a commutative ring. The canonical homomorphisms from ''R'' to the quotients <math>R/I^n</math> induce a homomorphism <math>R \to \hat{R}</math>. The latter homomorphism is injective if ''R'' is a noetherian integral domain and ''I'' is a proper ideal, or if ''R'' is a noetherian local ring with maximal ideal ''I'', by [[Krull's intersection theorem]].<ref>Atiyah and Macdonald, Theorem 10.17 and its corollaries.</ref> The construction is especially useful when ''I'' is a maximal ideal.
| |
| | |
| The basic example is the completion '''Z'''<sub>''p''</sub> of '''Z''' at the principal ideal (''p'') generated by a prime number ''p''; it is called the ring of [[p-adic integer|''p''-adic integers]]. The completion can in this case be constructed also from the [[p-adic absolute value|''p''-adic absolute value]] on '''Q'''. The [[p-adic absolute value|''p''-adic absolute value]] on '''Q''' is a map <math>x \mapsto |x|</math> from '''Q''' to '''R''' given by <math>|n|_p=p^{-v_p(n)}</math> where <math>v_p(n)</math> denotes the exponent of ''p'' in the prime factorization of a nonzero integer ''n'' into prime numbers (we also put <math>|0|_p=0</math> and <math>|m/n|_p = |m|_p/|n|_p</math>). It defines a distance function on '''Q''' and the completion of '''Q''' as a [[metric space]] is denoted by '''Q'''<sub>''p''</sub>. It is again a field since the field operations extend to the completion. The subring of '''Q'''<sub>''p''</sub> consisting of elements ''x'' with <math>|x|_p \le 1</math> is isomorphic to '''Z'''<sub>''p''</sub>.
| |
| | |
| Similarly, the formal power series ring <math>R[\![t]\!]</math> is the completion of <math>R[t]</math> at <math>(t)</math>.
| |
| | |
| See also: [[Hensel's lemma]].<!-- need to be discussed with concrete examples. Start of a para: The notion of the completion formalizes the construction of a solution by successive approximation. -->
| |
| | |
| A complete ring has much simpler structure than a commutative ring. This owns to the [[Cohen structure theorem]], which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the [[integral closure]] and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of [[excellent ring]].
| |
| | |
| ===Group ring ===
| |
| This construction allows one to study a group using the ring theory. Let ''G'' be a group and ''A'' a commutative ring. The [[group ring]] <math>A[G]</math> of ''G'' over ''A'' is then the set of all functions <math>f: G \to A</math> such that <math>f(s) = 0</math> for all but finitely many ''s'' in ''G'' with addition and multiplication defined as follows. Let <math>R = A[G]</math> and make it an abelian group with the ordinary addition of functions. The multiplication on it is given by [[convolution]]:
| |
| :<math>(f*g)(t) = \sum_{s \in G} f(s)g(s^{-1}t)</math>.
| |
| This is a finite sum and is therefore well-defined. Also, the function <math>f*g</math> belongs to <math>R</math>. One then checks that the addition and the multiplication satisfy the ring axioms. ''R'' has the multiplicative identity <math>\delta_1</math> where <math>\delta_t(t) = 1, \delta_t(s) = 0</math> for all <math>s \ne t</math>. (cf. [[Kronecker delta]].) <math>\delta_t, \, t \in G,</math> form an ''A''-basis of ''R''.<ref>In representation theory, when ''G'' is an abelian group, <math>\delta_s</math> is often denoted by <math>e^s</math>.</ref> Explicitly, for any ''f'' in ''R'', there is the expansion
| |
| :<math>f = \sum_{s \in G} f(s) \delta_s.</math>
| |
| | |
| Finally, essentially the same construction is possible for a unital semigroup instead of a group except the multiplication is given by:<ref name="Lang 2.3">{{harvnb|Lang|2002|loc=Ch II, §3}}</ref>
| |
| :<math>(f*g)(t) = \sum_{uv = t} f(u)g(v)</math>.
| |
| The resulting ring is called a [[semigroup ring]]. For example, <math>A[\mathbb{N}_0]</math> is a polynomial ring of one variable over ''A''.
| |
| | |
| ===Tensor product===
| |
| {{main|Tensor product of algebras}}
| |
| Let ''A'', ''B'' be algebras over a commutative ring ''R''. Then the tensor product of ''R''-modules <math>A \otimes_R B</math> is a ''R''-module. We can turn it to a ring by extending linearly <math>(x \otimes u) (y \otimes v) = xy \otimes uv</math>. For example, if ''R<nowiki>'</nowiki>'' is an ''R''-algebra, then <math>R' \otimes_R R[t] \simeq R'[t]</math> and <math>R[t] \otimes_R R[t] \simeq R[t_1, t_2]</math>. To give a nontrivial example, let <math>\epsilon, t</math> be indeterminates and view <math>R, R[\epsilon]</math> as algebras over <math>R[t]</math> via <math>t \mapsto 0, t \mapsto \epsilon^2</math>. One can show:<ref>{{harvnb|Eisenbud–Harris|2000|loc=Exercise I-46 (g)}}</ref>
| |
| :<math>R[\epsilon] \otimes_{R[t]} R \simeq R[\epsilon]/(\epsilon^2)</math>
| |
| (The ring on the right is called the ring of [[dual number]]s.)
| |
| | |
| If ''A'' is a ''k''-algebra, then <math>A_n \simeq A \otimes_k k_n</math>. We also have: for algebras ''A'', ''A' '' over ''k'' and their subalgebras ''B'', ''B''', resp.,
| |
| :<math>C_{A \otimes A'}(B \otimes B') = C_A(B) \otimes C_{A'}(B')</math>
| |
| where <math>C_A(B)</math> refers to the centralizer of ''B'' in ''A''.<ref name="harvnb|Milne|CFT|loc=Ch IV, §2">{{harvnb|Milne|CFT|loc=Ch IV, §2}}</ref> In particular, the center of <math>A \otimes B</math> is the tensor product of the centers of ''A'' and ''B''.
| |
| | |
| ===Rings with generators and relations===
| |
| The most general way to construct a ring is by specifying generators and relations. Let ''F'' be a [[free ring]] (i.e., free algebra over the integers) with the set ''X'' of symbols; i.e., ''F'' consists of polynomials with integral coefficients in noncommuting variables that are elements of ''X''. A free ring satisfies the universal property that one expects on a free object; that is, any function from the set ''X'' to a ring ''R'' factors through ''F'' so that <math>F \to R</math> is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.<ref>{{citation|title=Skew Fields: Theory of General Division Rings|volume=57|series=Encyclopedia of Mathematics and its Applications|first=Paul Moritz|last=Cohn|publisher=Cambridge University Press|year=1995|isbn=9780521432177|page=242|url=http://books.google.com/books?id=u-4ADgUgpSMC&pg=PA242}}.</ref>
| |
| | |
| Now, we can impose relations among symbols in ''X'' by taking a quotient. Explicitly, if ''E'' is a subset of ''F'', then the quotient ring of ''F'' by the ideal generated by ''E'' is called the ring with generators ''X'' and relations ''E''. If we used a ring, say, ''A'' as a base ring instead of '''Z''', then the resulting ring will be over ''A''. For example, if <math>E = \{ xy - yx \mid x, y \in X \}</math>, then the resulting ring will be the usual polynomial ring with coefficients in ''A'' in variables that are elements of ''X'' (It is also the same thing as the [[symmetric algebra]] over ''A'' with symbols ''X''.)
| |
| | |
| == Special kinds of rings ==
| |
| | |
| ===Domains===
| |
| A [[zero ring|nonzero]] ring with no nonzero [[zero-divisor]]s is called a [[domain (ring theory)|domain]]. A commutative domain is called an [[integral domain]]. The most important integral domains are principal ideals domains, PID for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a [[unique factorization domain]], an integral domain in which every nonunit element is a product of prime elements. (an element ''x'' is prime if <math>(x)</math> is a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the [[ring of integers]] (not necessarily rational integers) fails to be a PID.
| |
| | |
| Among theorems concerning a PID, the most important one is the [[structure theorem for finitely generated modules over a principal ideal domain]]. The theorem may be illustrated by the following application to linear algebra.<ref>{{harvnb|Lang|2002|loc=Ch XIV, §2}}</ref> Let ''V'' be a finite-dimensional vector space over a field ''k'' and <math>f: V \to V</math> a linear map with minimal polynomial ''q''. Then, since <math>k[t]</math> is a unique factorization domain, ''q'' factors into powers of distinct irreducible polynomials (i.e., prime elements):
| |
| :<math>q = p_1^{e_1} ... p_s^{e_s}.</math>
| |
| Letting <math>t \cdot v = f(v)</math>, we make ''V'' a ''k''[''t'']-module. The structure theorem then says that <math>V = \bigoplus V_i</math> as ''k''[''t'']-module where each <math>V_i</math> is isomorphic to a direct sum of submodules <math>W</math> isomorphic to <math>k[t]/(p_i^{k_j})</math>. Now, if <math>p_i(t) = t - \lambda_i</math>, then such a <math>W</math> has a basis in which the restriction of ''f'' is represented by a [[Jordan matrix]], Thus, if, say, ''k'' is algebraically closed, then <math>p_i</math> are of the form <math>t - \lambda_i</math> and the above decomposition corresponds to the [[Jordan canonical form]] of ''f''.
| |
| | |
| Any nonzero subring of a field is necessarily an integral domain. The converse is also true: an integral domain is always a subring of its [[field of fractions]]. This only partially generalizes to a noncommutative setting.
| |
| | |
| In algebraic geometry, UFD's arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a [[regular local ring]]. A regular local ring is a UFD.<ref>{{harvnb|Weibel|loc=Ch 1, Theorem 3.8}}</ref>
| |
| | |
| The following is a chain of [[subclass (set theory)|class inclusions]] that describes the relationship between rings, domains and fields:
| |
| | |
| * '''[[Commutative ring]]s''' ⊃ '''[[integral domain]]s''' ⊃ '''[[integrally closed domain]]s''' ⊃'''[[unique factorization domain]]s''' ⊃ '''[[principal ideal domain]]s''' ⊃ '''[[Euclidean domain]]s''' ⊃ '''[[field (mathematics)|fields]]'''
| |
| | |
| ===Division ring===
| |
| A [[division ring]] is a ring such that every non-zero element is a unit. A commutative division ring is a [[field (mathematics)|field]]. A prominent example of a division ring that is not a field is the ring of [[quaternion]]s. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every ''finite'' domain (in particular finite division ring) is a field; in particular commutative (the [[Wedderburn's little theorem]]).
| |
| | |
| The study of conjugacy classes figures prominently in the classical theory of division rings. Cartan famously asked the following question: given a division ring ''D'' and a proper sub-division-ring ''S'' that is not contained in the center, does each inner automorphism of ''D'' restrict to an automorphism of ''S''? The answer is negative: this is the [[Cartan–Brauer–Hua theorem]].
| |
| <!-- need more context. To give a concrete example, we consider a division ring ''D'' whose center ''k'' is the field <math>\mathbb{Q}_p</math> of ''p''-adic rational numbers (or more generally a nonarchimedean local field).<ref>{{harvnb|Milne|CFT|loc=Ch IV, §4}}</ref> The valuation of ''k'' uniquely extends to any subfield of ''D''. Since ''D'' is a union of subfields, we thus obtain the valuation ''v'' of ''D''. (This has to be verified; it is not a priori obvious that ''v'' is indeed a valuation). Define
| |
| :<math>O_D = \{ x \in D | v(x) \ge 0 \}, \quad \mathfrak{P} = \{ x \in D | v(x) > 0 \}</math>.
| |
| Then <math>O_D</math> is a subring of ''D'' with the unique maximal ideal that is <math>\mathfrak{P}</math>. It is called the [[ring of integers]] of ''D''.-->
| |
| | |
| ===Semisimple rings===
| |
| A ring is called a [[semisimple ring]] if it is semisimple as a left module (or right module) over itself. A ring is called a [[semiprimitive ring]] if its [[Jacobson radical]] is zero. (The Jacobson radical is the intersection of all maximal left ideals.) A ring is semisimple if and only if it is [[artinian ring|artinian]] and is semiprimitive.
| |
| | |
| An algebra over a field ''k'' is artinian if and only if it has finite dimension. Thus, a semisimple algebra over a field is necessarily finite-dimensional, while a simple algebra may have infinite-dimension; e.g., the [[ring of differential operators]].
| |
| | |
| Any module over a semisimple ring is semisimple. (Proof: any free module over a semisimple ring is clearly semisimple and any module is a quotient of a free module.)
| |
| | |
| Examples of semisimple rings:
| |
| *A matrix ring over a division ring is semisimple (actually simple).
| |
| *The group ring <math>k[G]</math> of a finite group ''G'' over a field ''k'' is semisimple if the characteristic of ''k'' does not divide the order of ''G''. ([[Maschke's theorem]])
| |
| * The [[Weyl algebra]] (over a field) is a simple ring; it is not semisimple since it has infinite dimension and thus not artinian.
| |
| *[[Clifford algebra]]s are semisimple.
| |
| | |
| Semisimplicity is closely related to separability. An [[algebra over a field|algebra]] ''A'' over a field ''k'' is said to be [[separable algebra|separable]] if the base extension <math>A \otimes_k F</math> is semisimple for any [[field extension]] <math>F/k</math>. If ''A'' happens to be a field, then this is equivalent to the usual definition in field theory (cf. [[separable extension]].)
| |
| | |
| ===Central simple algebra and Brauer group===
| |
| {{main|Central simple algebra}}
| |
| | |
| For a field ''k'', a ''k''-algebra is central if its center is ''k'' and is simple if it is a [[simple ring]]. Since the center of a simple ''k''-algebra is a field, any simple ''k''-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a ''k''-algebra. The matrix ring of size ''n'' over a ring ''R'' will be denoted by <math>R_n</math>.
| |
| | |
| The [[Skolem–Noether theorem]] states any automorphism of a central simple algebra is inner.
| |
| | |
| Two central simple algebras ''A'' and ''B'' are said to be ''similar'' if there are integers ''n'' and ''m'' such that <math>A \otimes_k k_n \approx B \otimes_k k_m</math>.<ref name="harvnb|Milne|CFT|loc=Ch IV, §2"/> Since <math>k_n \otimes_k k_m \simeq k_{nm}</math>, the similarity is an equivalence relation. The similarity classes <math>[A]</math> with the multiplication <math>[A][B] = [A \otimes_k B]</math> form an abelian group called the [[Brauer group]] of ''k'' and is denoted by <math>\operatorname{Br}(k)</math>. By the [[Artin–Wedderburn theorem]], a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring.
| |
| | |
| For example, <math>\operatorname{Br}(k)</math> is trivial if ''k'' is a finite field or an algebraically closed field (more generally [[quasi-algebraically closed field]]; cf. [[Tsen's theorem]]). <math>\operatorname{Br}(\mathbb{R})</math> has order 2 (a special case of the [[Frobenius theorem (real division algebras)|theorem of Frobenius]]). Finally, if ''k'' is a nonarchimedean [[local field]] (e.g., <math>\mathbb{Q}_p</math>), then <math>\operatorname{Br}(k) = \mathbb{Q}/\mathbb{Z}</math> through the [[Hasse invariant of an algebra|invariant map]].
| |
| | |
| Now, if ''F'' is a field extension of ''k'', then the base extension <math>- \otimes_k F</math> induces <math>\operatorname{Br}(k) \to \operatorname{Br}(F)</math>. Its kernel is denoted by <math>\operatorname{Br}(F/k)</math>. It consists of <math>[A]</math> such that <math>A \otimes_k F</math> is a matrix ring over ''F'' (i.e., ''A'' is split by ''F''.) If the extension is finite and Galois, then <math>\operatorname{Br}(F/k)</math> is canonically isomorphic to <math>H^2(\operatorname{Gal}(F/k), k^*)</math>.<ref>Serre, J-P ., Applications algébriques de la cohomologie des groupes, I, II, Séminaire Henri Cartan, 1950/51 [http://www.numdam.org/numdam-bin/feuilleter?id=SHC_1950-1951__3_]</ref>
| |
| | |
| [[Azumaya algebra]]s generalize the notion of central simple algebras to a commutative local ring.
| |
| | |
| == Rings with extra structure ==
| |
| | |
| A ring may be viewed as an [[abelian group]] (by using the addition operation), with extra structure. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
| |
| *An ''[[associative algebra]]'' is a ring that is also a [[vector space]] over a field ''K''. For instance, the set of ''n''-by-''n'' matrices over the real field '''R''' has dimension ''n''<sup>2</sup> as a real vector space.
| |
| *A ring ''R'' is a ''[[topological ring]]'' if its set of elements is given a [[topological space|topology]] which makes the addition map ( <math>+ : R\times R \to R\,</math>) and the multiplication map ( <math>\cdot : R\times R \to R\,</math>) to be both [[Continuous function (topology)|continuous]] as maps between topological spaces (where ''X'' × ''X'' inherits the [[product topology]] or any other product in the category). For example, ''n''-by-''n'' matrices over the real numbers could be given either the [[Euclidean topology]], or the [[Zariski topology]], and in either case one would obtain a topological ring.
| |
| | |
| == Some examples of the ubiquity of rings ==
| |
| | |
| Many different kinds of [[mathematical object]]s can be fruitfully analyzed in terms of some [[functor|associated ring]].
| |
| | |
| ===Cohomology ring of a topological space===
| |
| To any [[topological space]] ''X'' one can associate its integral [[cohomology ring]]
| |
| :<math>H^*(X,\mathbb{Z}) = \bigoplus_{i=0}^{\infty} H^i(X,\mathbb{Z}),</math>
| |
| a [[graded ring]]. There are also [[homology group]]s <math>H_i(X,\mathbb{Z})</math> of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the [[sphere]]s and [[torus|tori]], for which the methods of [[point-set topology]] are not well-suited. [[Cohomology group]]s were later defined in terms of homology groups in a way which is roughly analogous to the dual of a [[vector space]]. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the [[universal coefficient theorem]]. However, the advantage of the cohomology groups is that there is a [[cup product|natural product]], which is analogous to the observation that one can multiply pointwise a ''k''-[[multilinear form]] and an ''l''-multilinear form to get a (''k'' + ''l'')-multilinear form.
| |
| | |
| The ring structure in cohomology provides the foundation for [[characteristic class]]es of [[fiber bundle]]s, intersection theory on manifolds and [[algebraic variety|algebraic varieties]], [[Schubert calculus]] and much more.
| |
| | |
| ===Burnside ring of a group===
| |
| To any [[group (mathematics)|group]] is associated its [[Burnside ring]] which uses a ring to describe the various ways the group can [[group action|act]] on a finite set. The Burnside ring's additive group is the [[free abelian group]] whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the [[representation ring]]: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.
| |
| | |
| ===Representation ring of a group ring===
| |
| To any [[group ring]] or [[Hopf algebra]] is associated its [[representation ring]] or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from [[character theory]], which is more or less the [[Grothendieck group]] given a ring structure.
| |
| | |
| ===Function field of an irreducible algebraic variety===
| |
| To any irreducible [[algebraic variety]] is associated its [[function field of an algebraic variety|function field]]. The points of an algebraic variety correspond to [[valuation ring]]s contained in the function field and containing the [[coordinate ring]]. The study of [[algebraic geometry]] makes heavy use of [[commutative algebra]] to study geometric concepts in terms of ring-theoretic properties. [[Birational geometry]] studies maps between the subrings of the function field.
| |
| | |
| ===Face ring of a simplicial complex===
| |
| Every [[simplicial complex]] has an associated face ring, also called its [[Stanley–Reisner ring]]. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in [[algebraic combinatorics]]. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of [[simplicial polytope]]s.
| |
| | |
| == Category theoretical description ==
| |
| {{main|Category of rings}}
| |
| | |
| Every ring can be thought of as a [[monoid (category theory)|monoid]] in '''Ab''', the [[category of abelian groups]] (thought of as a [[monoidal category]] under the [[tensor product of abelian groups|tensor product of <math>{\mathbb Z}</math>-modules]]). The monoid action of a ring ''R'' on an abelian group is simply an [[module (mathematics)|''R''-module]]. Essentially, an ''R''-module is a generalization of the notion of a [[vector space]] – where rather than a vector space over a field, one has a "vector space over a ring".
| |
| | |
| Let (''A'', +) be an abelian group and let End(''A'') be its [[endomorphism ring]] (see above). Note that, essentially, End(''A'') is the set of all morphisms of ''A'', where if ''f'' is in End(''A''), and ''g'' is in End(''A''), the following rules may be used to compute ''f'' + ''g'' and ''f'' '''·''' ''g'':
| |
| | |
| * (''f'' + ''g'')(''x'') = ''f''(''x'') + ''g''(''x'')
| |
| | |
| * (''f'' '''·''' ''g'')(''x'') = ''f''(''g''(''x''))
| |
| | |
| where + as in ''f''(''x'') + ''g''(''x'') is addition in ''A'', and function composition is denoted from right to left. Therefore, [[functor|associated]] to any abelian group, is a ring. Conversely, given any ring, (''R'', +, '''·''' ), (''R'', +) is an abelian group. Furthermore, for every ''r'' in ''R'', right (or left) multiplication by ''r'' gives rise to a morphism of (''R'', +), by right (or left) distributivity. Let ''A'' = (''R'', +). Consider those [[endomorphism]]s of ''A'', that "factor through" right (or left) multiplication of ''R''. In other words, let End<sub>R</sub>(''A'') be the set of all morphisms ''m'' of ''A'', having the property that ''m''(''r'' '''·''' ''x'') = ''r'' '''·''' ''m''(''x''). It was seen that every ''r'' in ''R'' gives rise to a morphism of ''A'': right multiplication by ''r''. It is in fact true that this association of any element of ''R'', to a morphism of ''A'', as a function from ''R'' to End<sub>R</sub>(''A''), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian ''X''-group (by ''X''-group, it is meant a group with ''X'' being its [[Group with operators|set of operators]]).<ref>Jacobson (2009), p. 162, Theorem 3.2.</ref> In essence, the most general form of a ring, is the endomorphism group of some abelian ''X''-group.
| |
| | |
| Any ring can be seen as a [[preadditive category]] with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. [[Additive functor]]s between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of [[morphism]]s closed under addition and under composition with arbitrary morphisms.
| |
| | |
| == Generalization ==
| |
| | |
| Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms.
| |
| | |
| ===Rng===
| |
| A [[rng (algebra)|rng]] is the same as a ring, except that the existence of a [[multiplicative identity]] is not assumed.<ref>Jacobson 2009.</ref>
| |
| | |
| ===Nonassociative ring===
| |
| A [[nonassociative ring]] is an algebraic structure that satisfies all of the ring axioms but the [[associative property|associativity]] and the existence of a [[multiplicative identity]]. A notable example is a [[Lie algebra]]. There exists some structure theory<!-- developed by Jacobson? --> for such algebras that generalizes the analogous results for Lie algebras and associative algebras.{{citation needed|date=November 2013}}
| |
| | |
| ===Semiring===
| |
| A [[semiring]] is obtained by weakening the assumption that (''R'',+) is an abelian group to the assumption that (''R'',+) is a commutative monoid, and adding the axiom that 0 · ''a'' = ''a'' · 0 = 0 for all ''a'' in ''R'' (since it no longer follows from the other axioms).
| |
| | |
| == Other ring-like objects ==
| |
| | |
| ===Ring object in a category===
| |
| Let ''C'' be a category with finite products. Let pt denote a [[terminal object]] of ''C'' (an empty product). A '''ring object''' in ''C'' is an object ''R'' equipped with morphisms <math>R \times R \stackrel{a}\to R</math> (addition), <math>R \times R \stackrel{m}\to R</math> (multiplication), <math>\operatorname{pt} \stackrel{0}\to R</math> (additive identity), <math>R \stackrel{i}\to R</math> (additive inverse), and <math>\operatorname{pt} \stackrel{1}\to R</math> (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object ''R'' equipped with a factorization of its functor of points <math>h_R = \operatorname{Hom}(-,R) : C^{\operatorname{op}} \to \mathbf{Sets}</math> through the category of rings: <math>C^{\operatorname{op}} \to \mathbf{Rings} \stackrel{\textrm{forgetful}}\longrightarrow \mathbf{Sets}</math>.
| |
| | |
| ===Ring scheme===
| |
| In algebraic geometry, a '''ring scheme''' over a base [[scheme (mathematics)|scheme]] ''S'' is a ring object in the category of ''S''-schemes. One example is the ring scheme W<sub>''n''</sub> over Spec '''Z''', which for any commutative ring ''A'' returns the ring W<sub>''n''</sub>(''A'') of ''p''-isotypic Witt vectors of length ''n'' over ''A''.<ref>Serre, p. 44.</ref>
| |
| | |
| ===Ring spectrum===
| |
| In [[algebraic topology]], a [[ring spectrum]] is a [[spectrum (topology)|spectrum]] ''X'' together with a multiplication <math>\mu \colon X \wedge X \to X</math> and a unit map <math>S \to X</math> from the [[sphere spectrum]] ''S'', such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a [[monoid object]] in a good category of spectra such as the category of [[symmetric spectrum|symmetric spectra]].
| |
| | |
| == See also ==
| |
| | |
| {{Wikibooks|Abstract Algebra/Rings}}
| |
| <div style="-moz-column-count:2; column-count:2;">
| |
| * [[Algebra over a commutative ring]]
| |
| * [[Algebraic structure]]
| |
| * [[Category of rings]]
| |
| * [[Chinese remainder theorem]]
| |
| * [[Glossary of ring theory]]
| |
| * [[Nonassociative ring]]
| |
| * [[Ring theory]]
| |
| * [[Semiring]]
| |
| * [[Spectrum of a ring]]
| |
| * [[Simplicial commutative ring]]
| |
| * Special types of rings:
| |
| ** [[Boolean ring]]
| |
| ** [[Commutative ring]]
| |
| ** [[Dedekind ring]]
| |
| ** [[Differential ring]]
| |
| ** [[Division ring]] (skew field)
| |
| ** [[Exponential field|Exponential ring]]
| |
| ** [[Field (mathematics)|Field]]
| |
| ** [[Integral domain]]
| |
| ** [[Lie ring]]
| |
| ** [[Local ring]]
| |
| ** [[Noetherian ring|Noetherian]] and [[artinian ring]]s
| |
| ** [[Ordered ring]]
| |
| ** [[Principal ideal domain]] (PID)
| |
| ** [[Reduced ring]]
| |
| ** [[Regular ring]]
| |
| ** [[Ring of periods]]
| |
| ** [[Ring theory]]
| |
| ** [[SBI ring]]
| |
| ** [[Unique factorization domain]] (UFD)
| |
| ** [[Valuation ring]] and [[discrete valuation ring]]
| |
| ** [[Zero ring]]
| |
| </div>
| |
| | |
| == Notes ==
| |
| | |
| {{cnote|a|Some authors only require that a ring be a [[semigroup]] under multiplication; that is, do not require that there be a multiplicative identity (1). See the section [[#Notes on the definition|Notes on the definition]] for more details.}}
| |
| {{cnote|b|Elements which do have multiplicative inverses are called [[unit (ring theory)|units]], see {{Harvard citations|nb = yes|last = Lang|year = 2002|loc =§II.1, p. 84}}.}}
| |
| {{cnote|c|The closure axiom is already implied by the condition that +/• be a binary operation. Some authors therefore omit this axiom. {{Harvard citations|nb = yes|last = Lang|year = 2002}}}}
| |
| {{cnote|d|The transition from the integers to the rationals by adding fractions is generalized by the [[quotient field]].}}
| |
| {{cnote|e|Many authors include [[commutative ring|commutativity of rings]] in the set of ''ring axioms'' ([[Ring (mathematics)#Formal definition|see above]]) and therefore refer to "commutative rings" as just "rings".}}
| |
| | |
| ===Citations===
| |
| {{reflist|3}}
| |
| | |
| == References ==
| |
| | |
| ===General references===
| |
| * {{Cite book
| |
| | last=Artin
| |
| | first=Michael
| |
| | author-link=Michael Artin
| |
| | title=Algebra
| |
| | publisher=Prentice-Hall
| |
| | year=1991
| |
| }}
| |
| * {{Cite book
| |
| | last1=Atiyah
| |
| | first1=Michael
| |
| | author1-link=Michael Atiyah
| |
| | last2=Macdonald
| |
| | first2=Ian G.
| |
| | author2-link=Ian G. Macdonald
| |
| | title=Introduction to commutative algebra
| |
| | publisher=Addison–Wesley
| |
| | year=1969
| |
| }}
| |
| * {{Cite book
| |
| | last1=Bourbaki
| |
| | first1=N.
| |
| | author1-link=Nicolas Bourbaki
| |
| | title=Algebra I, Chapters 1-3
| |
| | publisher=Springer
| |
| | year=1998
| |
| }}
| |
| * {{Citation
| |
| | last1=Cohn
| |
| | first1=Paul Moritz
| |
| | title=Basic algebra: groups, rings, and fields
| |
| | year=2003
| |
| | publisher=Springer
| |
| | isbn=978-1-85233-587-8
| |
| }}.
| |
| * {{Cite book
| |
| | last=Eisenbud
| |
| | first=David
| |
| | author-link=David Eisenbud
| |
| | title=Commutative algebra with a view toward algebraic geometry
| |
| | publisher=Springer
| |
| | year=1995
| |
| }}
| |
| * {{Cite book
| |
| | last=Herstein
| |
| | first=I. N.
| |
| | author-link=Israel Nathan Herstein
| |
| | others=With an afterword by Lance W. Small
| |
| | title=Noncommutative rings
| |
| | series=Carus Mathematical Monographs
| |
| | volume=15
| |
| | publisher=Mathematical Association of America
| |
| | year=1994
| |
| | origyear=reprint of the 1968 original
| |
| | isbn=0-88385-015-X
| |
| }}
| |
| * {{Cite book
| |
| | title=Radical Theory of Rings
| |
| | publisher= Chapman & Hall/CRC Pure and Applied Mathematics
| |
| | first1=J.W.
| |
| | last1=Gardner
| |
| | first2=R.
| |
| | last2=Wiegandt
| |
| | year=2003
| |
| | isbn=0824750330
| |
| }}
| |
| * {{Cite book
| |
| | last=Jacobson
| |
| | first=Nathan
| |
| | author-link=Nathan Jacobson
| |
| | title=Basic algebra
| |
| | edition=2nd
| |
| | volume=1
| |
| | publisher=Dover
| |
| | year=2009
| |
| | isbn=978-0-486-47189-1
| |
| }}
| |
| * {{Cite journal
| |
| | last=Jacobson
| |
| | first=Nathan
| |
| | author-link=Nathan Jacobson
| |
| | title=Structure of rings
| |
| | journal=American Mathematical Society Colloquium Publications
| |
| | volume=37
| |
| | edition=Revised
| |
| | year=1964
| |
| }}
| |
| * {{Cite journal
| |
| | last=Jacobson
| |
| | first=Nathan
| |
| | author-link=Nathan Jacobson
| |
| | title=The Theory of Rings
| |
| | journal=American Mathematical Society Mathematical Surveys
| |
| | volume=I
| |
| | year=1943
| |
| }}
| |
| * {{Citation
| |
| | last1=Kaplansky
| |
| | first1=Irving
| |
| | author1-link=Irving Kaplansky
| |
| | title=Commutative rings
| |
| | publisher=[[University of Chicago Press]]
| |
| | edition=Revised
| |
| | mr=0345945
| |
| | year=1974
| |
| | isbn=0-226-42454-5
| |
| }}.
| |
| * {{Cite book
| |
| | last=Lam
| |
| | first=Tsit Yuen
| |
| | author-link=Tsit Yuen Lam
| |
| | title=A first course in noncommutative rings
| |
| | edition=2nd
| |
| | series=Graduate Texts in Mathematics
| |
| | volume=131
| |
| | publisher=Springer
| |
| | year=2001
| |
| | isbn=0-387-95183-0
| |
| }}
| |
| * {{Cite book
| |
| | last=Lam
| |
| | first=Tsit Yuen
| |
| | author-link=Tsit Yuen Lam
| |
| | title=Exercises in classical ring theory
| |
| | edition=2nd
| |
| | series=Problem Books in Mathematics
| |
| | publisher=Springer
| |
| | year=2003
| |
| | isbn=0-387-00500-5
| |
| }}
| |
| * {{Cite book
| |
| | last=Lam
| |
| | first=Tsit Yuen
| |
| | author-link=Tsit Yuen Lam
| |
| | title=Lectures on modules and rings
| |
| | series=Graduate Texts in Mathematics
| |
| | volume=189
| |
| | publisher=Springer
| |
| | year=1999
| |
| | isbn=0-387-98428-3
| |
| }}
| |
| * {{Lang Algebra|edition=3r}}.
| |
| * {{Cite book
| |
| | last1=Matsumura
| |
| | first1=Hideyuki
| |
| | title=Commutative Ring Theory
| |
| | publisher=[[Cambridge University Press]]
| |
| | edition=2nd
| |
| | series=Cambridge Studies in Advanced Mathematics
| |
| | year=1989
| |
| | isbn=978-0-521-36764-6
| |
| }}
| |
| * {{Cite web
| |
| | last=Milne
| |
| | first=J.
| |
| | title=A primer of commutative algebra
| |
| | url=http://www.jmilne.org/math/xnotes/ca.html
| |
| }}
| |
| * {{Citation
| |
| | last1=Rotman
| |
| | first1=Joseph
| |
| | title=Galois Theory
| |
| | publisher=Springer
| |
| | edition=2nd
| |
| | year=1998
| |
| | isbn=0-387-98541-7
| |
| }}.
| |
| * {{Citation
| |
| | last1=van der Waerden
| |
| | first1=Bartel Leendert
| |
| | author1-link=Bartel Leendert van der Waerden
| |
| | title=[[Moderne Algebra]]. Teil I
| |
| | publisher=Springer
| |
| | series= Die Grundlehren der mathematischen Wissenschaften
| |
| | volume=33
| |
| | isbn=978-3-540-56799-8
| |
| | id={{MR|0009016}}{{MR|0037277}}{{MR|0069787}}{{MR|0122834}}{{MR|0177027}}{{MR|0263581}}
| |
| | year=1930
| |
| }}.
| |
| * {{Cite book
| |
| | first1=Raymond Louis
| |
| | last1=Wilder
| |
| | title=Introduction to Foundations of Mathematics
| |
| | publisher=Wiley
| |
| | year=1965
| |
| }}
| |
| * {{Cite book
| |
| | last1=Zariski
| |
| | first1=Oscar
| |
| | last2=Samuel
| |
| | first2=Pierre
| |
| | title=Commutative Algebra
| |
| | volume=1
| |
| | publisher=Van Nostrand
| |
| | year=1958
| |
| }}
| |
| | |
| ===Special references===
| |
| * {{Citation
| |
| | last1=Balcerzyk
| |
| | first1=Stanisław
| |
| | last2=Józefiak
| |
| | first2=Tadeusz
| |
| | title=Commutative Noetherian and Krull rings
| |
| | publisher=Ellis Horwood Ltd.
| |
| | location=Chichester
| |
| | series=Mathematics and its Applications
| |
| | year=1989
| |
| | isbn=978-0-13-155615-7
| |
| }}.
| |
| * {{Citation
| |
| | last1=Balcerzyk
| |
| | first1=Stanisław
| |
| | last2=Józefiak
| |
| | first2=Tadeusz
| |
| | title=Dimension, multiplicity and homological methods
| |
| | publisher=Ellis Horwood Ltd.
| |
| | location=Chichester
| |
| | series=Mathematics and its Applications
| |
| | isbn=978-0-13-155623-2
| |
| | year=1989
| |
| }}.
| |
| * {{Cite journal
| |
| | last=Ballieu
| |
| | first=R.
| |
| | title=Anneaux finis; systèmes hypercomplexes de rang trois sur un corps commutatif
| |
| | journal=Ann. Soc. Sci. Bruxelles
| |
| | volume=I
| |
| | issue=61
| |
| | pages=222–227
| |
| | year=1947
| |
| }}
| |
| * {{Cite book
| |
| | last1=Berrick
| |
| | first1=A. J.
| |
| | last2=Keating
| |
| | first2=M. E.
| |
| | title=An Introduction to Rings and Modules with K-Theory in View
| |
| | publisher=Cambridge University Press
| |
| | year=2000
| |
| }}
| |
| * {{Citation
| |
| | last1=Eisenbud
| |
| | first1=David
| |
| | author1-link=David Eisenbud
| |
| | title=Commutative algebra. With a view toward algebraic geometry.
| |
| | publisher=Springer
| |
| | series=Graduate Texts in Mathematics
| |
| | volume=150
| |
| | mr=1322960
| |
| | year=1995
| |
| | isbn=978-0-387-94268-1
| |
| }}.
| |
| * {{Cite journal
| |
| | last1=Gilmer
| |
| | first1=R.
| |
| | last2=Mott
| |
| | first2=J.
| |
| | title=Associative Rings of Order
| |
| | journal=Proc. Japan Acad.
| |
| | volume=49
| |
| | pages=795–799
| |
| | year=1973
| |
| }}
| |
| * {{Cite book
| |
| | last1=Harris
| |
| | first1=J. W.
| |
| | last2=Stocker
| |
| | first2=H.
| |
| | title=Handbook of Mathematics and Computational Science
| |
| | publisher=Springer
| |
| | year=1998
| |
| }}
| |
| * {{Citation
| |
| | last1=Jacobson
| |
| | first1=Nathan
| |
| | author1-link=Nathan Jacobson
| |
| | title=Structure theory of algebraic algebras of bounded degree
| |
| | journal=[[Annals of Mathematics]]
| |
| | issn=0003-486X
| |
| | volume=46
| |
| | issue=4
| |
| | pages=695–707
| |
| | doi=10.2307/1969205
| |
| | jstor=1969205
| |
| | publisher=Annals of Mathematics
| |
| | year=1945
| |
| }}.
| |
| * {{Cite book
| |
| | last=Knuth
| |
| | first=D. E.
| |
| | author-link=Donald Knuth
| |
| | title=The Art of Computer Programming
| |
| | volume=Vol. 2: Seminumerical Algorithms
| |
| | edition=3rd
| |
| | publisher=Addison–Wesley
| |
| | year=1998
| |
| }}
| |
| * {{Cite book
| |
| | last1=Korn
| |
| | first1=G. A.
| |
| | last2=Korn
| |
| | first2=T. M.
| |
| | title=Mathematical Handbook for Scientists and Engineers
| |
| | publisher=Dover
| |
| | year=2000
| |
| }}
| |
| * {{Cite web
| |
| | last=Milne
| |
| | first=J.
| |
| | title=Class field theory
| |
| | url=http://www.jmilne.org/math/CourseNotes/cft.html
| |
| }}
| |
| * {{Citation
| |
| | last1=Nagata
| |
| | first1=Masayoshi
| |
| | author1-link=Masayoshi Nagata
| |
| | title=Local rings
| |
| | publisher=Interscience Publishers
| |
| | series=Interscience Tracts in Pure and Applied Mathematics
| |
| | year=1962
| |
| | origyear=1975 reprint
| |
| | mr=0155856
| |
| | volume=13
| |
| | isbn=978-0-88275-228-0
| |
| }}.
| |
| * {{Cite book
| |
| | last=Pierce
| |
| | first=Richard S.
| |
| | title=Associative algebras
| |
| | series=Graduate Texts in Mathematics
| |
| | volume=88
| |
| | publisher=Springer
| |
| | year=1982
| |
| | isbn=0-387-90693-2
| |
| }}
| |
| * {{Citation
| |
| | last=Serre
| |
| | first=Jean-Pierre
| |
| | author1-link=Jean-Pierre Serre
| |
| | title=Local fields
| |
| | series=Graduate Texts in Mathematics
| |
| | volume=67
| |
| | publisher=Springer
| |
| | year=1979
| |
| }}.
| |
| * {{Citation
| |
| | last=Springer
| |
| | first=Tonny A.
| |
| | title=Invariant theory
| |
| | series=Lecture Notes in Mathematics
| |
| | volume=585
| |
| | publisher=Springer
| |
| | year=1977
| |
| }}.
| |
| * {{Cite web
| |
| | last=Weibel
| |
| | first=Charles
| |
| | title=The K-book: An introduction to algebraic K-theory
| |
| | url=http://www.math.rutgers.edu/~weibel/Kbook.html
| |
| }}
| |
| * {{Cite book
| |
| | last1=Zariski
| |
| | first1=Oscar
| |
| | author1-link=Oscar Zariski
| |
| | last2=Samuel
| |
| | first2=Pierre
| |
| | author2-link=Pierre Samuel
| |
| | title=Commutative algebra
| |
| | series=Graduate Texts in Mathematics
| |
| | volume=28-29
| |
| | publisher=Springer
| |
| | year=1975
| |
| | isbn=0-387-90089-6
| |
| }}
| |
| | |
| ===Primary sources===
| |
| * {{Cite journal
| |
| | last=Fraenkel
| |
| | first=A.
| |
| | author-link=Abraham Fraenkel
| |
| | title=Über die Teiler der Null und die Zerlegung von Ringen
| |
| | journal=J. reine angew. Math.
| |
| | volume=145
| |
| | pages=139–176
| |
| | year=1914
| |
| }}
| |
| * {{Cite journal
| |
| | last=Hilbert
| |
| | first=David
| |
| | author-link=David Hilbert
| |
| | title=Die Theorie der algebraischen Zahlkörper
| |
| | journal=Jahresbericht der Deutschen Mathematiker Vereinigung
| |
| | volume=4
| |
| | year=1897
| |
| }}
| |
| * {{Cite journal
| |
| | last=Noether
| |
| | first=Emmy
| |
| | author-link=Emmy Noether
| |
| | title=Idealtheorie in Ringbereichen
| |
| | journal=Math. Annalen
| |
| | volume=83
| |
| | pages=24–66
| |
| | year=1921
| |
| }}
| |
| | |
| ===Historical references===
| |
| *[http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Ring_theory.html History of ring theory at the MacTutor Archive]
| |
| * Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillian, 1996.
| |
| * Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 4th ed. New York: Springer-Verlag, 2004. ISBN 3-540-43491-7.
| |
| * Faith, Carl, ''Rings and things and a fine array of twentieth century associative algebra''. Mathematical Surveys and Monographs, 65. American Mathematical Society, Providence, RI, 1999. xxxiv+422 pp. ISBN 0-8218-0993-8.
| |
| * Itô, K. (Ed.). "Rings." §368 in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, 1986.
| |
| * Kleiner, I., "The Genesis of the Abstract Ring Concept", Amer. Math. Monthly 103, 417–424, 1996.
| |
| * Kleiner, I., "From numbers to rings: the early history of ring theory", ''Elem. Math.'' '''53''' (1998), 18–35.
| |
| * Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24–34, 2005.
| |
| * Singmaster, D. and Bloom, D. M. "Problem E1648." Amer. Math. Monthly 71, 918–920, 1964.
| |
| * Van der Waerden, B. L. A History of Algebra. New York: Springer-Verlag, 1985.
| |
| | |
| {{DEFAULTSORT:Ring (Mathematics)}}
| |
| [[Category:Algebraic structures]]
| |
| [[Category:Mathematical structures]]
| |
| [[Category:Ring theory]]
| |