## Definition ## A __unital $\mathbb{Z}$-algebra__ is an [[Z-algebra|$\mathbb{Z}$-algebra]] $(A, +, -, 0, \cdot)$ with * a term $1: A$ * a left unit identity for $\cdot$ $$u_\lambda:\prod_{(a:G)} 1 \cdot a = a$$ * a right unit identity for $\cdot$ $$u_\rho:\prod_{(a:G)} a \cdot 1 = a$$ ## Properties ## A unital $\mathbb{Z}$-algebra is an [[H-space]] in [[abelian group]]s. Every untial $\mathbb{Z}$-algebra where $0$ has a two-sided inverse is [[contractible]]. ## Examples ## * Every [[contractible type]] is a unital $\mathbb{Z}$-algebra. * The [[integers]] are a unital $\mathbb{Z}$-algebra. * The [[rational numbers]] are a unital $\mathbb{Z}$-algebra. ## See also ## * [[H-space]] * [[Z-algebra]] * [[ring]] * [[power function]] * [[polynomial function]]