## Definiton ## Let $A$ be an [[abelian group]], let $R$ be a [[commutative ring]]. $A$ is an $R$-module if it comes with an [[abelian group homomorphism]] $\alpha:R \to (A \to A)$ such that * $\alpha(1) = id_A$ * for all $a:R$ and $b:R$, $\alpha(a) \circ \alpha(b) = \alpha(a \cdot b)$ ## Properties ## Every abelian group is a $\mathbb{Z}$-module. ## See also ## * [[abelian group]] * [[graded module]]