Bimodule - Knowlpedia

Let $R,S$ be rings. An $(R,S)$ -bimodule is an abelian group $M$ that is simultaneously a left module over $R$ and a right module over $S$ , such that the actions are compatible: for all $r\in R$ , $s\in S$ , and $m\in M$ , one has $(rm)s=r(ms)$ .

Bimodules are the natural setting for many constructions (e.g. balanced products) and are the input for the tensor product, where compatibility of left/right actions is essential.

Examples

Any ring $R$ is an $(R,R)$ -bimodule with actions given by multiplication on the left and right.
If $M$ is a left $R$ -module, then $M$ is an $(R,\mathbb Z)$ -bimodule using the canonical right $\mathbb Z$ -action coming from the abelian-group structure.
If $A$ is an algebra over a ring $R$ , then $A$ is naturally an $(R,A)$ -bimodule: $r\cdot a$ uses the structure map $R\to A$ , and $a\cdot a'$ is multiplication in $A$ .