division ring | skew field

A division algebra, also called a "division ring" or "skew field, " is a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative. Every field is therefore also a division algebra. In French, the term "corps non commutatif" is used to mean division algebra, while "corps" alone means field. Explicitly, a division algebra is a set together with two binary operators (S, + , *) satisfying the following conditions: 1. Additive associativity: For all a, b, c element S, (a + b) + c = a + (b + c). 2. Additive commutativity: For all a, b element S, a + b = b + a.

alternative algebra | Cayley number | field | group | Jordan algebra | Lie algebra | nonassociative algebra | power associative algebra | quaternion | Schur's lemma | unit ring | zero divisor