## Definition ## A __discrete reciprocal ring__ is a [[ring]] $(A, +, -, 0, \cdot, 1)$ with a commutative identity for $\cdot$: $$m_\kappa:\prod_{(a:A)} \prod_{(b:A)} a\cdot b = b\cdot a$$ ## Properties ## Every commutative discrete reciprocal ring is a [[commutative discrete division ring]]. ## Examples ## * The [[rational numbers]] are a commutative discrete reciprocal ring. * Every [[commutative discrete division ring]] is a commutative discrete reciprocal ring. * Every [[discrete field]] is a commutative discrete reciprocal ring. ## See also ## * [[commutative ring]] * [[commutative reciprocal ring]]