Definition:Bimodule

Definition
Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {S, +_S, \times_S}$ be a ring.

A bimodule over $\paren{R,S}$ is an abelian group $\struct {G, +_G}$ with binary operations:
 * $\circ_R : R \times G \to G$
 * $\circ_S : G \times S \to G$

such that:
 * $\quad \struct {G, +_G, \circ_R}$ is a left module
 * $\quad \struct {G, +_G, \circ_S}$ is a right module
 * $\quad \forall \lambda \in R: \forall \mu \in S: \forall x \in G : \paren{\lambda \circ_R x} \circ_S \mu = \lambda \circ_R \paren {x \circ_S \mu}$

If $\struct {S, +_S, \times_S} = \struct {R, +_R, \times_R}$ then a bimodule over $\paren{R,R}$ is simply called a bimodule over $R$