198.102.153.1 at 23:47, 26 November 2014

2014-11-26T23:47:22Z

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en>Incnis Mrsi: /* Examples */ see talk: Division ring #the_only_non-trivial_automorphism_of_complex_numbers

2014-02-17T06:40:32Z

Examples: see talk: Division ring #the_only_non-trivial_automorphism_of_complex_numbers

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en>Ebony Jackson: trivial ring --> zero ring

2013-12-18T01:13:08Z

trivial ring --> zero ring

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In [[abstract algebra]], a '''division ring''', also called a '''skew field''', is a [[ring (mathematics)|ring]] in which [[division (mathematics)|division]] is possible. Specifically, it is a [[zero ring|nonzero]] ring<ref>In this article, rings have a 1.</ref> in which every nonzero element ''a'' has a [[multiplicative inverse]], i.e., an element ''x'' with {{nowrap|1=''a''·''x'' = ''x''·''a'' = 1}}. Stated differently, a ring is a division ring if and only if the [[group of units]] equals the set of all nonzero elements.

Division rings differ from [[field (mathematics)|fields]] only in that their multiplication is not required to be [[commutative]]. However, by [[Wedderburn's little theorem]] all finite division rings are commutative and therefore [[finite field]]s. Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.

==Relation to fields and linear algebra==
All fields are division rings; more interesting examples are the non-commutative division rings. The best known example is the ring of [[quaternion]]s '''H'''. If we allow only [[rational number|rational]] instead of [[real number|real]] coefficients in the constructions of the quaternions, we obtain another division ring. In general, if ''R'' is a ring and ''S'' is a [[simple module]] over ''R'', then, by [[Schur's lemma]], the [[endomorphism ring]] of ''S'' is a division ring;<ref>Lam (2001), {{Google books quote|id=f15FyZuZ3-4C|page=33|text=Schur's Lemma|p. 33}}.</ref> every division ring arises in this fashion from some simple module.

Much of [[linear algebra]] may be formulated, and remains correct, for (left) [[module (mathematics)|modules]] over division rings instead of [[vector space]]s over fields. Every module over a division ring has a basis; linear maps between finite-dimensional modules over a division ring can be described by [[matrix (mathematics)|matrices]], and the [[Gaussian elimination]] algorithm remains applicable. Differences between linear algebra over fields and skew fields occur whenever the order of the factors in a product matters. For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.

The [[center of a ring|center]] of a division ring is commutative and therefore a field.<ref>Simple commutative rings are fields. See Lam (2001), {{Google books quote|id=f15FyZuZ3-4C|page=39|text=simple commutative rings|p. 39}} and {{Google books quote|id=f15FyZuZ3-4C|page=45|text=center of a simple ring|exercise 3.4 on p.45}}.</ref> Every division ring is therefore a [[division algebra]] over its center. Division rings can be roughly classified according to whether or not they are finite-dimensional or infinite-dimensional over their centers. The former are called ''centrally finite'' and the latter ''centrally infinite''. Every field is, of course, one-dimensional over its center. The ring of [[Hamiltonian quaternions]] forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.

==Examples==
* As noted above, all [[Field (mathematics)|fields]] are division rings.
* The [[Real number|real]] and [[Rational number|rational]] [[quaternion]]s are strictly noncommutative division rings.
* Let <math>\sigma:\mathbb{C}\rightarrow\mathbb{C}</math> be a nontrivial [[automorphism]] of the field <math>\mathbb{C}</math> onto itself (''e.g.'', [[Complex conjugate|complex conjugation]]). Let <math>\mathbb{C}((z,\sigma))</math> denote the [[Ring of formal Laurent series|ring of]] [[formal Laurent series]] with complex coefficients, wherein multiplication is defined as follows: instead of simply allowing coefficients to commute directly with the indeterminate <math>z</math>, for <math>\alpha\in\mathbb{C}</math>, define <math>z^i\alpha:=\sigma^i(\alpha)z^i</math> for each index <math>i\in\mathbb{Z}</math>. The resulting ring of Laurent series is a strictly noncommutative division ring known as a ''skew Laurent series ring''.<ref>Lam (2001), p. 10</ref> This concept can be generalized to the ring of Laurent series over any fixed field <math>F</math>, given a nontrivial <math>F</math>-automorphism <math>\sigma</math>.

==Ring theorems==
'''[[Wedderburn's little theorem]]''': All finite division rings are commutative and therefore [[finite field]]s. ([[Ernst Witt]] gave a simple proof.)

'''[[Frobenius theorem (real division algebras)|Frobenius theorem]]''': The only finite-dimensional associative division algebras over the reals are the reals themselves, the [[complex number]]s, and the [[quaternion]]s.

==Related notions==
Division rings ''used to be'' called "fields" in an older usage. In many languages, a word meaning "body" is used for division rings, in some languages designating either commutative or non-commutative division rings, while in others specifically designating commutative division rings (what we now call fields in English). A more complete comparison is found in the article [[Field (mathematics)]].

Skew fields have an interesting [[lexical semantics|semantic]] feature: a modifier (here "skew") ''widens'' the scope of the base term (here "field"). Thus a field is a particular type of skew field, and not all skew fields are fields.

While division rings and algebras as discussed here are assumed to have associative multiplication, [[Division algebra#Not necessarily associative division algebras|nonassociative division algebras]] such as the [[octonion]]s are also of interest.

A [[near-field (mathematics)|near-field]] is an algebraic structure similar to a division ring, except that it has only one of the two [[distributive law]]s.

== Notes ==
<references />

==See also==
*[[Hua's identity]]

== References ==
* {{cite book |last1=Lam |first1=Tsit-Yuen |authorlink1= |last2= |first2= |authorlink2= |title=A first course in noncommutative rings |url= |edition=2 |series=Graduate texts in mathematics |volume=131 |year=2001 |publisher=Springer |location= |isbn=0-387-95183-0 |id= }}

==External links==
*[http://planetmath.org/?op=getobj&from=objects&id=3627 Proof of Wedderburn's Theorem at Planet Math]

[[Category:Ring theory]]

Division ring - Revision history

198.102.153.1 at 23:47, 26 November 2014

en>Incnis Mrsi: /* Examples */ see talk: Division ring #the_only_non-trivial_automorphism_of_complex_numbers

en>Ebony Jackson: trivial ring --> zero ring