## Definiton ## ### A single action ring Let $R$ be a [[ring]]. An $R$-bimodule is an [[abelian group]] $B$ with a [[trilinear function|trilinear]] multiplicative [[biaction|$R$-biaction]] $(-)(-)(-):R \times B \times R \to B$. ### Two different action rings Let $R$ and $S$ be [[ring]]s. A $R$-$S$-bimodule is an [[abelian group]] $B$ with a [[trilinear function|trilinear]] multiplicative [[biaction|$R$-$S$-biaction]] $(-)(-)(-):R \times B \times S \to B$. ## 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]] * [[trilinear function]] * [[biaction]] * [[module]] * [[algebra (module theory)]] * [[bimodule homomorphism]] * [[ideal (ring theory)]] * [[vector space]]