Division Ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units is the set of all non-zero elements.

Division rings differ from 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 fields. Historically, division rings were sometimes referred to as fields, while fields were called “commutative fields”.

Read more about Division RingRelation To Fields and Linear Algebra, Examples, Ring Theorems, Related Notions

Other articles related to "ring, division ring, rings, division rings, division":

Schur's Lemma - Formulation in The Language of Modules
... If M and N are two simple modules over a ring R, then any homomorphism f M → N of R-modules is either invertible or zero ... In particular, the endomorphism ring of a simple module is a division ring ... since any representation of a group G can equivalently be viewed as a module over the group ring of G ...
Artin–Wedderburn Theorem
... algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras ... The theorem states that an (Artinian) semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to ... In particular, any simple left or right Artinian ring is isomorphic to an n-by-n matrix ring over a division ring D, where both n and D are uniquely determined ...
Division Ring - 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 division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonions are also of interest ...
Simple Ring - Wedderburn's Theorem
... Wedderburn's theorem characterizes simple rings with a unit and a minimal left ideal ... is a generalization of the second assumption.) Namely it says that every such ring is, up to isomorphism, a ring of n × n matrices over a division ring ... Let D be a division ring and M(n,D) be the ring of matrices with entries in D ...
Ring (mathematics) - Basic Concepts - Division Ring
... A division ring is a ring such that every non-zero element is a unit ... A commutative division ring is called a field ... A main example of a division ring is the ring of quaternions ...

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