## Definiton ## Let $A$ be an [[abelian group]], let $R$ be a [[commutative ring]], and let $\alpha_l:R \times A \to A$ be a left multiplicative $R$-[[action]] on $A$ and $\alpha_r:A \times R \to A$ be a right multiplicative $R$-[[action]] on $A$. $A$ is a __left $R$-module__ if $\alpha_l$ is a [[bilinear function]], and $A$ is a __right $R$-module__ if $\alpha_r$ is a bilinear function. ## Properties ## Every abelian group is a left $\mathbb{Z}$-module and a right $\mathbb{Z}$-module. ## See also ## * [[abelian group]] * [[action]] * [[monoid]] * [[algebra (ring theory)]]