Division ring - Definition and Overview

In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1). Division rings are very similar to fields except that their multiplication is not required to be commutative. The condition 0 ≠ 1 is only there to exclude the trivial ring with a single element 0 = 1.

All fields are division rings. A more typical example is given by the quaternions. If we allow only rational instead of 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 the endomorphism ring of S is a division ring; every division ring arises in this fashion from some simple module.

All of linear algebra may be formulated, and remains correct, for modules over division rings instead of vector spaces 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 matrices, and the Gauss-Jordan elimination algorithm remains applicable.

Wedderburn's theorem states that all finite division rings are commutative and therefore finite fields.