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| | Pearl powder is a finely milled powder from quality freshwater pearls and its naturally compatible and easily absorbed by the skin and body. L-Glutathione pills have the ability to lighten dark pigmentations on the skin such as scars, dark underarms, freckles, etc. One study suggests a novel method of testing multiple topical skin-lightening methods prior to initiating therapy, using UV-induced skin tanning. They're very natural, so lemons are a great way of bleaching the skin and helping to reduce the appearance of your acne scars. Using sandalwood powder, lemon juice, tomato juice and cucumber juice, make a paste. <br><br>Bleaching cream can also reduce scars not just your skin pigmentation. Less commonly, some Black women will develop a decrease in melanin or postinflammatory hypopigmentation in response to skin trauma (burns, etc. Try and look for a product which contains Alpha Arbutin, as this substance has had a very high success rate when whitening a variety of skin tones. It is highly important to know which was the cause which led to the apparition of your dark patches or brown spots in order to know how to treat them. Europe has banned skin bleaching creams that contain hydroquinone. <br><br>Research thoroughly all products associated with skin lightening by reading up on each product including what's in it, how it actually works to solve your skin problems, adverse effects as well as the maintaining aspects and other qualities of the products. If acne is constantly bringing you down, you probably need to change a few things in your routine. In fact, for many people, it is recommended that you shower twice daily to ensure freshness and to avoid odor. According to these colors and to the type of skin you have certain products are recommended and certain are prohibited. They work simply by ramping way up mobile or portable return costs. <br><br>Cucumber has cooling effects, and it has the same amount of hydrogen as your skin. Essentially, any dark colored area of the skin can be considered hyperpigmentation � age spots, acne scars, melasma, uneven skin tone, sun damage, etc. Even those with fair skin want to keep their skin tone as white as possible, and Asians are fond of whiter skin so they use all kinds of products to skin lighten and whiten much lighter than their natural skin tone. Cucumber juice is also an excellent bleaching agent. When it comes to pet first aid, hydrogen peroxide might also be of use. <br><br>This amazing ingredient is a powerful antioxidant that also helps stimulate the production of new healthy skin cells. But don't hold your breadth; even if it's diagnosed as strongyloides stercoralis, doctors are very unfamiliar with its treatment. That's right, no need to apply anything else to the skin, just the honey. Melanocytes treated with kojic acid become nondendritic with a decreased melanin content. In the time of applying Yogurt on the skin for about ten minutes to a minimum of thirty minutes, the skin is going to be expected to turn into lighter and more vibrant and this really is recognized by many to include a great deal of lactic acid.<br><br>If you have any queries pertaining to exactly where and how to use [http://stylax.info/ how to bleach skin], you can contact us at our site. |
| [[Ring theory]] is the branch of [[mathematics]] in which [[ring (mathematics)|rings]] are studied: that is, structures supporting both an [[addition]] and a [[multiplication]] operation. This is a glossary of some terms of the subject.
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| ==Definition of a ring==
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| ;'''[[Ring (mathematics)|ring]]''' : A ''ring'' is a [[Set (mathematics)|set]] ''R'' with two [[binary operation]]s, usually called addition (+) and multiplication (*), such that ''R'' is an [[abelian group]] under addition, ''R'' is a [[monoid]] under multiplication, and multiplication is both left and right [[distributive]] over addition. Rings are assumed to have multiplicative identities unless otherwise noted. The additive identity is denoted by 0 and the multiplicative identity by 1. (''Warning'': some books, especially older books, use the term "ring" to mean what here will be called a [[rng (algebra)|rng]]; i.e., they do not require a ring to have a multiplicative identity.)
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| ; '''[[subring]]''' : A subset ''S'' of the ring (''R'',+,*) which remains a ring when + and * are restricted to ''S'' and contains the multiplicative identity 1 of ''R'' is called a ''subring'' of ''R''.
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| ==Types of elements==
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| ; '''[[Central element|central]]''' : An element ''r'' of a ring ''R'' is ''central'' if ''xr'' = ''rx'' for all ''x'' in ''R''. The set of all central elements forms a [[subring]] of ''R'', known as the ''center'' of ''R''.
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| ; '''[[divisor]]''' : In an [[integral domain]] ''R'', an element ''a'' is called a ''divisor'' of the element ''b'' (and we say ''a'' ''divides'' ''b'') if there exists an element ''x'' in ''R'' with ''ax'' = ''b''.
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| ; '''[[Idempotent element|idempotent]]''' : An element ''r'' of a ring is ''idempotent'' if ''r''<sup>2</sup> = ''r''.
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| ; '''[[integral element]]''': For a commutative ring ''B'' containing a subring ''A'', an element ''b'' is ''integral over A'' if it satisfies a monic polynomial with coefficients from ''A''.
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| ; '''[[Irreducible element|irreducible]]''' : An element ''x'' of an integral domain is ''irreducible'' if it is not a unit and for any elements ''a'' and ''b'' such that ''x''=''ab'', either ''a'' or ''b'' is a unit. Note that every prime element is irreducible, but not necessarily vice versa.
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| ; '''[[prime element]]''' : An element ''x'' of an integral domain is a ''prime element'' if it is not zero and not a unit and whenever ''x'' divides a product ''ab'', ''x'' divides ''a'' or ''x'' divides ''b''.
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| ; '''[[Primordial element (algebra)|primordial element]]'''
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| ; '''[[nilpotent]]''' : An element ''r'' of ''R'' is ''nilpotent'' if there exists a positive integer ''n'' such that ''r''<sup>''n''</sup> = 0.
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| ; '''[[Unit (ring theory)|unit]]''' or '''invertible element''' : An element ''r'' of the ring ''R'' is a ''unit'' if there exists an element ''r''<sup>−1</sup> such that ''rr''<sup>−1</sup>=''r''<sup>−1</sup>''r''=1. This element ''r''<sup>−1</sup> is uniquely determined by ''r'' and is called the ''multiplicative inverse'' of ''r''. The set of units forms a [[group (mathematics)|group]] under multiplication.
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| ; '''[[zero divisor]]''' : A nonzero element ''r'' of ''R'' is a ''zero divisor'' if there exists a nonzero element ''s'' in R such that ''sr'' = 0 or ''rs'' = 0. Some authors opt to include zero as a zero divisor.
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| ==Homomorphisms and ideals==
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| ; '''[[factor ring]]''' or '''quotient ring''' : Given a ring ''R'' and an ideal ''I'' of ''R'', the ''factor ring'' is the ring formed by the set ''R''/''I'' of [[coset]]s {''a+I'' : ''a''∈''R''} together with the operations (''a+I'')+(''b+I'')=(''a''+''b'')+''I'' and (''a+I'')(''b+I'')=''ab+I''. The relationship between ideals, homomorphisms, and factor rings is summed up in the [[fundamental theorem on homomorphisms]].
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| ; '''finitely generated ideal''' : A left ideal ''I'' is ''finitely generated'' if there exist finitely many elements ''a''<sub>1</sub>,...,''a''<sub>''n''</sub> such that ''I'' = ''Ra''<sub>1</sub> + ... + ''Ra''<sub>''n''</sub>. A right ideal ''I'' is ''finitely generated'' if there exist finitely many elements ''a''<sub>1</sub>,...,''a''<sub>''n''</sub> such that ''I'' = ''a''<sub>1</sub>''R'' + ... + ''a''<sub>''n''</sub>''R''. A two-sided ideal ''I'' is ''finitely generated'' if there exist finitely many elements ''a''<sub>1</sub>,...,''a''<sub>''n''</sub> such that ''I'' = ''Ra''<sub>1</sub>''R'' + ... + ''Ra''<sub>''n''</sub>''R''.
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| ; '''[[Ideal (ring theory)|ideal]]''' : A ''left ideal'' ''I'' of ''R'' is a subgroup of ''R'' such that ''aI'' ⊆ ''I'' for all ''a''∈''R''. A ''right ideal'' is a subgroup of ''R'' such that ''Ia''⊆''I'' for all ''a''∈''R''. An ''ideal'' (sometimes called a ''two-sided ideal'' for emphasis) is a subgroup which is both a left ideal and a right ideal.
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| ; '''[[Jacobson radical]]''' : The intersection of all maximal left ideals in a ring forms a two-sided ideal, the ''Jacobson radical'' of the ring.
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| ; '''[[kernel (algebra)|kernel]] of a ring homomorphism''' : The ''kernel'' of a ring homomorphism ''f'' : ''R'' → ''S'' is the set of all elements ''x'' of ''R'' such that ''f''(''x'') = 0. Every ideal is the kernel of a ring homomorphism and vice versa.
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| ; '''[[maximal ideal]]''' : A left ideal ''M'' of the ring ''R'' is a ''maximal left ideal'' if ''M'' ≠ ''R'' and the only left ideals containing ''M'' are ''R'' and ''M'' itself. Maximal right ideals are defined similarly. In commutative rings, there is no difference, and one speaks simply of ''maximal ideals''.
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| ; '''[[nil ideal]]''' : An ideal is ''nil'' if it consists only of nilpotent elements.
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| ; '''[[nilpotent ideal]]''' : An ideal ''I'' is ''nilpotent'' if the [[product of ideals|power]] ''I''<sup>''k''</sup> is {0} for some positive integer ''k''. Every nilpotent ideal is nil, but the converse is not true in general.
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| ; '''[[Nilradical of a ring|nilradical]]''' : The set of all nilpotent elements in a commutative ring forms an ideal, the ''nilradical'' of the ring. The nilradical is equal to the intersection of all the ring's [[prime ideal]]s. It is contained in, but in general not equal to, the ring's Jacobson Radical.
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| ; '''[[prime ideal]]''' : An ideal ''P'' in a [[commutative ring]] ''R'' is ''prime'' if ''P'' ≠ ''R'' and if for all ''a'' and ''b'' in ''R'' with ''ab'' in ''P'', we have ''a'' in ''P'' or ''b'' in ''P''. Every maximal ideal in a commutative ring is prime. There is also a definition of prime ideal for noncommutative rings.
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| ; '''[[principal ideal]]''' : A ''principal left ideal'' in a ring ''R'' is a left ideal of the form ''Ra'' for some element ''a'' of ''R''. A ''principal right ideal'' is a right ideal of the form ''aR'' for some element ''a'' of ''R''. A ''principal ideal'' is a two-sided ideal of the form ''RaR'' for some element ''a'' of ''R''.
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| ; '''[[radical of an ideal]]''' : The radical of an ideal ''I'' in a [[commutative ring]] consists of all those ring elements a power of which lies in ''I''. It is equal to the intersection of all prime ideals containing ''I''.
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| ; '''[[ring homomorphism]]''' : A [[function (mathematics)|function]] ''f'' : ''R'' → ''S'' between rings (''R'',+,*) and (''S'',⊕,×) is a ''ring homomorphism'' if it satisfies
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| :: ''f''(''a'' + ''b'') = ''f''(''a'') ⊕ ''f''(''b'')
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| :: ''f''(''a'' * ''b'') = ''f''(''a'') × ''f''(''b'')
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| :: ''f''(1) = 1
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| :for all elements ''a'' and ''b'' of ''R''.
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| ; '''[[ring monomorphism]]''' : A ring homomorphism that is [[injective]] is a ''ring monomorphism''.
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| ; '''[[ring isomorphism]]''' : A ring homomorphism that is [[bijective]] is a ''ring isomorphism''. The inverse of a ring isomorphism is also a ring isomorphism. Two rings are ''isomorphic'' if there exists a ring isomorphism between them. Isomorphic rings can be thought as essentially the same, only with different labels on the individual elements.
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| ; '''trivial ideal''': Every nonzero ring ''R'' is guaranteed to have two ideals: the zero ideal and the entire ring ''R''. These ideals are usually referred to as ''trivial ideals''. Right ideals, left ideals, and two-sided ideals other than these are called ''nontrivial''.
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| ==Types of rings==
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| ; '''Abelian ring''' : A ring in which all [[idempotent element]]s are [[Center (algebra)|central]] is called an Abelian ring. Such rings need not be commutative.
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| ; '''[[artinian ring]]''' : A ring satisfying the [[descending chain condition]] for left ideals is ''left artinian''; if it satisfies the descending chain condition for right ideals, it is ''right artinian''; if it is both left and right artinian, it is called ''artinian''. Artinian rings are noetherian.
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| ; '''[[boolean ring]]''' : A ring in which every element is multiplicatively [[idempotent element|idempotent]] is a ''boolean ring''.
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| ; '''[[commutative ring]]''' : A ring ''R'' is ''commutative'' if the multiplication is commutative, i.e. ''rs''=''sr'' for all ''r'',''s''∈''R''.
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| ; '''[[Dedekind domain]]''' : A ''Dedekind domain'' is an integral domain in which every ideal has a unique factorization into prime ideals.
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| ; '''[[division ring]]''' or '''skew field''' : A ring in which every nonzero element is a unit and 1≠0 is a ''division ring''.
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| ; '''[[Domain (ring theory)|domain]]''' : A ''domain'' is a nonzero ring with no zero divisors except 0. This is the noncommutative generalization of [[integral domain]].
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| ; '''[[Euclidean domain]]''' : A ''Euclidean domain'' is an integral domain in which a [[degree function]] is defined so that "division with remainder" can be carried out. It is so named because the [[Euclidean algorithm]] is a well-defined algorithm in these rings. All Euclidean domains are principal ideal domains.
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| ; '''[[Field (mathematics)|field]]''' : A ''field'' is a commutative division ring. Every finite division ring is a field, as is every finite integral domain.
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| ; '''[[finitely generated ring]]''': a ring that is finitely generated as '''Z'''-algebra.
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| ; '''Finitely presented algebra'''<span id="Finitely presented algebra"></span>: If ''R'' is a [[commutative ring]] and ''A'' is an [[R-algebra|''R''-algebra]], then ''A'' is a '''finitely presented ''R''-algebra''' if it is a [[quotient ring|quotient]] of a [[polynomial ring]] over ''R'' in finitely many variables by a [[finitely generated ideal]].<ref>{{harvnb|Grothendieck|Dieudonné|1964|loc=§1.4.1}}</ref>
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| ;'''[[hereditary ring]]''': A ring is ''left hereditary'' if its left ideals are all projective modules. Right hereditary rings are defined analogously.
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| ; '''[[integral domain]]''' or '''entire ring''' : A nonzero [[commutative ring]] with no [[zero divisor]]s except 0.
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| ;'''[[invariant basis number]]''': A ring ''R'' has ''invariant basis number'' if ''R''<sup>''m''</sup> isomorphic to ''R''<sup>''n''</sup> as [[module (mathematics)|''R''-modules]] implies ''m''=''n''.
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| ; '''[[local ring]]''' : A ring with a unique maximal left ideal is a ''local ring''. These rings also have a unique maximal right ideal, and the left and the right unique maximal ideals coincide. Certain commutative rings can be embedded in local rings via [[localization of a ring|localization]] at a [[prime ideal]].
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| ; '''[[Noetherian ring]]''' : A ring satisfying the [[ascending chain condition]] for left ideals is ''left Noetherian''; a ring satisfying the ascending chain condition for right ideals is ''right Noetherian''; a ring that is both left and right Noetherian is ''Noetherian''. A ring is left Noetherian if and only if all its left ideals are finitely generated; analogously for right Noetherian rings.
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| ; '''null ring''': See [[rng (algebra)#Rng_of_square_zero|rng of square zero]].
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| ; '''[[perfect ring]]''': A ''left perfect ring'' is one satisfying the [[descending chain condition]] on ''right'' principal ideals. They are also characterized as rings whose flat left modules are all projective modules. Right perfect rings are defined analogously. Artinian rings are perfect.
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| ; '''[[prime ring]]''' : A [[zero ring|nonzero ring]] ''R'' is called a ''prime ring'' if for any two elements ''a'' and ''b'' of ''R'' with ''aRb'' = 0, we have either ''a = 0'' or ''b = 0''. This is equivalent to saying that the zero ideal is a prime ideal. Every [[simple ring]] and every [[domain (ring theory)|domain]] is a prime ring.
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| ; '''[[primitive ring]]''' : A ''left primitive ring'' is a ring that has a [[faithful module|faithful]] [[simple module|simple]] [[module (mathematics)|left ''R''-module]]. Every [[simple ring]] is primitive. Primitive rings are [[prime ring|prime]].
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| ; '''[[principal ideal domain]]''' : An integral domain in which every ideal is principal is a ''principal ideal domain''. All principal ideal domains are unique factorization domains.
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| ; '''[[quasi-Frobenius ring]]''' : a special type of Artinian ring which is also a [[self-injective ring]] on both sides. Every semisimple ring is quasi-Frobenius.
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| ; '''[[rng (algebra)#Rng_of_square_zero|rng of square zero]]''': A [[rng (algebra)|rng]] in which ''xy''=0 for all ''x'' and ''y''. These are sometimes also called '''zero rings''', even though they usually do not have a 1.
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| ; '''[[self-injective ring]]''': A ring ''R'' is left self-injective'' if the module <sub>''R''</sub>''R'' is an [[injective module]]. While rings with unity are always projective as modules, they are not always injective as modules.
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| ; '''[[semiprimitive ring]]''' or '''Jacobson semisimple ring''': This is a ring whose [[Jacobson radical]] is zero. Von Neumann regular rings and primitive rings are semiprimitive, however quasi-Frobenius rings and local rings are usually not semiprimitive.
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| ; '''[[semisimple ring]]''' : A ''semisimple ring'' is a ring ''R'' that has a "nice" decomposition, in the sense that ''R'' is a [[semisimple module|semisimple]] left ''R''-module. Every semisimple ring is also Artinian, and has no nilpotent ideals. The [[Artin–Wedderburn theorem]] asserts that every semisimple ring is a finite product of full matrix rings over division rings.
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| ; '''[[simple ring]]''' : A non-zero ring which only has trivial two-sided ideals (the zero ideal, the ring itself, and no more) is a ''simple ring''.
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| ; '''[[trivial ring]]''': The ring consisting only of a single element 0 = 1, also called the [[zero ring]].
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| ; '''[[unique factorization domain]]''' or '''factorial ring''': An integral domain ''R'' in which every non-zero non-[[Unit (ring theory)|unit]] element can be written as a product of [[prime element]]s of ''R''. This essentially means that every non-zero non-unit can be written uniquely as a product of irreducible elements. | |
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| ; '''[[von Neumann regular ring]]''': A ring for which each element ''a'' can be expressed as ''a''=''axa'' for another element ''x'' in the ring. Semisimple rings are von Neumann regular.
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| ; '''[[zero ring]]''': The ring consisting only of a single element 0 = 1, also called the [[trivial ring]]. Sometimes "zero ring" is alternatively used to mean [[rng (algebra)#Rng_of_square_zero|rng of square zero]].
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| ==Ring constructions==
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| ; '''[[direct product]]''' of a family of rings : This is a way to construct a new ring from given rings by taking the [[cartesian product]] of the given rings and defining the algebraic operations component-wise.
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| ; '''[[endomorphism ring]]''' : A ring formed by the [[endomorphism]]s of an algebraic structure. Usually its multiplication is taken to be [[function composition]], while its addition is pointwise addition of the images.
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| ; '''[[localization of a ring]]''' : For commutative rings, a technique to turn a given set of elements of a ring into units. It is named ''Localization'' because it can be used to make any given ring into a ''local'' ring. To localize a ring ''R'', take a multiplicatively closed subset ''S'' containing no [[zero divisor]]s, and formally define their multiplicative inverses, which shall be added into ''R''. Localization in noncommutative rings is more complicated, and has been in defined several different ways.
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| ;'''[[matrix ring]]''': Given a ring ''R'', it is possible to construct ''matrix rings'' whose entries come from ''R''. Often these are the square matrix rings, but under certain conditions "infinite matrix rings" are also possible. Square matrix rings arise as endomorphism rings of free modules with finite rank.
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| ; [[opposite ring]]
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| : Given a ring ''R'', its opposite ring <math>R^{\textrm{op}}</math> has the same underlying set as ''R'', the addition operation is defined as in ''R'', but the product of ''s'' and ''r'' in <math>R^{\textrm{op}}</math> is ''rs'', while the product is ''sr'' in ''R''.
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| ===Polynomial rings===
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| {{Main|Polynomial ring}}
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| ; [[Polynomial ring#Differential and skew-polynomial rings|differential polynomial ring]]
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| ; [[formal power series]] ring
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| ; [[Laurent polynomial]] ring
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| ; [[monoid ring]]
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| ; [[polynomial ring]]
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| : Given ''R'' a commutative ring. The polynomial ring ''R''[''x''] is defined to be the set <math> \{ a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_1 x + a_0 | a_n , a_{n-1}, a_{n-2}, \ldots , a_0 \in R \}</math> with addition defined by <math>\left(\sum_i a_i x^i\right) + \left(\sum_i b_i x^i\right) =
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| \sum_i (a_i+b_i)x^i</math>, and with multiplication defined by <math>\left(\sum_i a_i x^i\right) \cdot \left(\sum_j b_j x^j\right) =
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| \sum_k \left(\sum_{i,j: i + j = k} a_i b_j\right)x ^k</math>.
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| : Some results about properties of ''R'' and ''R''[''x'']:
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| :* If ''R'' is UFD, so is ''R''[''x''].
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| :* If ''R'' is Noetherian, so is ''R''[''x''].
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| ; ring of [[rational function]]s
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| ; [[Polynomial ring#Differential and skew-polynomial rings|skew polynomial ring]]
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| : Given ''R'' a ring and an endomorphism <math>\sigma \in \textrm{End}(R)</math> of ''R''. The skew polynomial ring <math>R[x; \sigma]</math> is defined to be the set <math> \{ a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_1 x + a_0 | a_n , a_{n-1}, a_{n-2}, \ldots , a_0 \in R \}</math>, with addition defined as usual, and multiplication defined by the relation <math>xa = \sigma(a) x \;\forall a \in R</math>.
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| ==Miscellaneous==
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| ; '''[[Characteristic (algebra)|characteristic]]''' : The ''characteristic'' of a ring is the smallest positive integer ''n'' satisfying ''nx'' = 0 for all elements ''x'' of the ring, if such an ''n'' exists. Otherwise, the characteristic is 0.
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| ; '''[[Krull dimension]] of a commutative ring''' : The maximal length of a strictly increasing chain of prime ideals in the ring.
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| ==Ringlike structures==
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| The following structures include generalizations and other algebraic objects similar to rings.
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| ; '''[[nearring]]''': A structure that is a group under addition, a [[semigroup]] under multiplication, and whose multiplication distributes on the right over addition.
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| ; '''[[Rng (algebra)|rng]]''' (or '''[[pseudo-ring]]'''): An algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element. The term "rng" is meant to suggest that it is a "r'''i'''ng" without an "'''i'''dentity".
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| ; '''[[semiring]]''' : An algebraic structure satisfying the same properties as a ring, except that addition need only be an abelian [[monoid]] operation, rather than an abelian group operation. That is, elements in a semiring need not have additive inverses.
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| ==See also==
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| * [[Glossary of field theory]]
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| * [[Glossary of module theory]]
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| * [[Glossary of commutative algebra]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{EGA | book=IV-1}}
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| {{DEFAULTSORT:Glossary Of Ring Theory}}
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| [[Category:Glossaries of mathematics|Ring theory]]
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| [[Category:Ring theory| ]]
| |
Pearl powder is a finely milled powder from quality freshwater pearls and its naturally compatible and easily absorbed by the skin and body. L-Glutathione pills have the ability to lighten dark pigmentations on the skin such as scars, dark underarms, freckles, etc. One study suggests a novel method of testing multiple topical skin-lightening methods prior to initiating therapy, using UV-induced skin tanning. They're very natural, so lemons are a great way of bleaching the skin and helping to reduce the appearance of your acne scars. Using sandalwood powder, lemon juice, tomato juice and cucumber juice, make a paste.
Bleaching cream can also reduce scars not just your skin pigmentation. Less commonly, some Black women will develop a decrease in melanin or postinflammatory hypopigmentation in response to skin trauma (burns, etc. Try and look for a product which contains Alpha Arbutin, as this substance has had a very high success rate when whitening a variety of skin tones. It is highly important to know which was the cause which led to the apparition of your dark patches or brown spots in order to know how to treat them. Europe has banned skin bleaching creams that contain hydroquinone.
Research thoroughly all products associated with skin lightening by reading up on each product including what's in it, how it actually works to solve your skin problems, adverse effects as well as the maintaining aspects and other qualities of the products. If acne is constantly bringing you down, you probably need to change a few things in your routine. In fact, for many people, it is recommended that you shower twice daily to ensure freshness and to avoid odor. According to these colors and to the type of skin you have certain products are recommended and certain are prohibited. They work simply by ramping way up mobile or portable return costs.
Cucumber has cooling effects, and it has the same amount of hydrogen as your skin. Essentially, any dark colored area of the skin can be considered hyperpigmentation � age spots, acne scars, melasma, uneven skin tone, sun damage, etc. Even those with fair skin want to keep their skin tone as white as possible, and Asians are fond of whiter skin so they use all kinds of products to skin lighten and whiten much lighter than their natural skin tone. Cucumber juice is also an excellent bleaching agent. When it comes to pet first aid, hydrogen peroxide might also be of use.
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