Bimodule

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 $A$.