## Definition ## A __$\mathbb{Z}$-algebra__ is an [[abelian group]] $(A, +, -, 0)$ with * a binary operation $(-)\cdot(-): A \times A \to A$ * a left distributive identity $$d_\lambda:\prod_{(a:G)} \prod_{(b:G)} \prod_{(c:G)} a \cdot (b + c) = a \cdot b + a \cdot c$$ * a right distributive identity $$d_\rho:\prod_{(a:G)} \prod_{(b:G)} \prod_{(c:G)} (a + b) \cdot c = a \cdot c + b \cdot c$$ ## Examples ## * Every [[contractible type]] is a $\mathbb{Z}$-algebra. * The [[integers]] are a $\mathbb{Z}$-algebra. * The [[rational numbers]] are a $\mathbb{Z}$-algebra. ## See also ## * [[abelian group]] * [[unital Z-algebra]] * [[cancellation Z-algebra]] * [[division Z-algebra]]