## Definition ## A __discrete division ring__ is a ring $(A, +, -, 0, \cdot)$ with * a discrete left divisibility identity $$d_\lambda:\prod_{(a:A)} ((a = 0) \to \emptyset) \times \prod_{(c:A)} \left\Vert \sum_{(b:A)} a \cdot b = c \right\Vert$$ * a discrete right divisibility identity $$d_\rho:\prod_{(a:A)} ((a = 0) \to \emptyset) \times \prod_{(c:A)} \left\Vert \sum_{(b:A)} b \cdot a = c \right\Vert$$ ## Properties ## Every discrete division ring is a [[discrete reciprocal ring]]. ## Examples ## * The [[rational numbers]] are a discrete division ring. * Every [[discrete reciprocal ring]] is a discrete division ring, and thus every [[discrete skewfield]] is a discrete division ring. ## See also ## * [[ring]] * [[division ring]]