## Definition ## A __discrete reciprocal ring__ is a [[ring]] $(A, +, -, 0, \cdot, 1)$ with * an identity showing that every term not equal to $0$ has a reciprocal (a two-sided multiplicative inverse) $$r:\prod_{(a:A)} \left( ((a = 0) \to \emptyset) \times \left\Vert \sum_{(b:A)} (a \cdot b = 1) \times (b \cdot a = 1) \right\Vert \right)$$ ## Properties ## Every discrete reciprocal ring is a [[discrete division ring]]. ## Examples ## * The [[rational numbers]] are a discrete reciprocal ring. * Every [[discrete division ring]] is a discrete reciprocal ring. * Every [[discrete skewfield]] is a discrete reciprocal ring. ## See also ## * [[ring]] * [[reciprocal ring]]