What is "division ring"

n. (context algebra English) A ring with 0 ≠ 1, such that every non-zero element a has a multiplicative inverse, meaning an element x with ax = xa = 1.

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with . Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements. A division ring is a type of noncommutative ring.

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”.