# Definition:Bimodule

## Definition

Let $\struct {R, +_R, \times_R}$ and $\struct {S, +_S, \times_S}$ be rings.

Let $\struct {G, +_G}$ be an abelian group.

Let $\circ_R : R \times G \to G$ and $\circ_S : G \times S \to G$ be binary operations such that:

$(1): \quad \struct {G, +_G, \circ_R}$ is a left module
$(2): \quad \struct {G, +_G, \circ_S}$ is a right module
$(3): \quad \forall \lambda \in R: \forall \mu \in S: \forall x \in G: \paren {\lambda \circ_R x} \circ_S \mu = \lambda \circ_R \paren {x \circ_S \mu}$

Then $\struct {G, +_G, \circ_R, \circ_S}$ is a bimodule over $\tuple {R, S}$.

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

## Also known as

A bimodule over $\tuple {R, S}$ can also be referred to as an $\tuple {R, S}$-bimodule.