## Definiton ## Let $R$ and $S$ be [[commutative ring]]s and let $M$ be a left $R$-module and a right $S$-module, with a left multiplicative [[bilinear function|bilinear]] $R$-[[action]] $\alpha_l:R \times A \to A$ and a right multiplicative [[bilinear function|bilinear]] $S$-[[action]] $\alpha_r:A \times S \to A$. $M$ is a __$R$-$S$-bimodule__ if $$p: \prod_{m:M} \prod_{a:R} \prod_{b:S} \alpha_l(a, \alpha_r(m, b)) = \alpha_r(\alpha_l(a, m), b)$$ For a commutative ring $R$, a $R$-$R$-bimodule is also called a __$R$-bimodule__. ## Properties ## * Every abelian group is a $\mathbb{Z}$-$\mathbb{Z}$-bimodule. * Every left $R$-module is a $R$-$\mathbb{Z}$-bimodule. * Every right $R$-module is a $\mathbb{Z}$-$R$-bimodule. ## See also ## * [[abelian group]] * [[module]] * [[algebra (module theory)]] * [[ideal (ring theory)]] * [[vector space]]