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^–1 a = 1. So, (right) division may be defined as a / b = a b^–1, but this notation is avoided, as one may have a b^–1 ≠ b^–1 a.

A commutative division ring is a field. Wedderburn's little theorem asserts that 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".^[7] In some languages, such as French, the word equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" (commutative field) or "corps gauche" (skew field).

All division rings are simple. That is, they have no two-sided ideal besides the zero ideal and itself.