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