< [[nlab:division algebra]] ## Definition ## A __discrete division $\mathbb{Z}$-algebra__ is a [[Z-algebra|$\mathbb{Z}$-algebra]] $(A, +, -, 0, \cdot)$ 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 $$d_\rho:\prod_{(a:A)} ((a = 0) \to \emptyset) \times \prod_{(c:A)} \left\Vert \sum_{(b:A)} b \cdot a = c \right\Vert$$ ## Examples ## * The [[rational numbers]] are a discrete division $\mathbb{Z}$-algebra. * Every [[discrete division ring]] is a discrete division $\mathbb{Z}$-algebra. ## See also ## * [[Z-algebra]] * [[division Z-algebra]]