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”.
Other articles related to "ring, division ring, rings, division rings, division":
... 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 ...
... 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 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 ...
... 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 ...
... 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 ...
Famous quotes containing the words ring and/or division:
“There is no magic decoding ring that will help us read our young adolescents feelings. Rather, what we need to do is hold out our antennae in the hope that well pick up the right signals.”
—The Lions Clubs International and the Quest Nation. The Surprising Years, III, ch.4 (1985)
“That crazed girl improvising her music,
Her poetry, dancing upon the shore,
Her soul in division from itself
Climbing, falling she knew not where,
Hiding amid the cargo of a steamship
Her knee-cap broken.”
—William Butler Yeats (18651939)