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Division ring. In algebra, a division ring, also called a skew field (or, occasionally, a sfield[1][2]), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring [3] in which every nonzero element a has a multiplicative inverse, that is, an element usually denoted a–1, such that a a–1 = a ...
In algebra, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings (group rings, division ...
Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field. The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.
A division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring [2] in which every nonzero 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 its group of units is the set of all nonzero elements.
Wedderburn's little theorem. In mathematics, Wedderburn's little theorem states that every finite division ring is a field. In other words, for finite rings, there is no distinction between domains, division rings and fields. The Artin–Zorn theorem generalizes the theorem to alternative rings: every finite alternative division ring is a field.
Unit (ring theory) In mathematics, element with a multiplicative inverse. In algebra, a unit or invertible element[a] of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that where 1 is the multiplicative identity; the element v is unique for this ...
Ring: a semiring whose additive monoid is an abelian group. Division ring: a nontrivial ring in which division by nonzero elements is defined. Commutative ring: a ring in which the multiplication operation is commutative. Field: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element).
The field of fractions of an integral domain is sometimes denoted by or (), and the construction is sometimes also called the fraction field, field of quotients, or quotient field of . All four are in common usage, but are not to be confused with the quotient of a ring by an ideal , which is a quite different concept.