Definition 13

Let $(A_\lambda)_{\lambda \in L},\ (B_\mu)_{\mu \in M}$ be two families of rings; an $((A_\lambda),\ (B_\mu))$*-multimodule* (or multimodule over the families of rings $(A_\lambda)_{\lambda \in L},\ (B_\mu)_{\mu \in M}$) is a set E with, for each $\lambda \in L$, a left $A_\lambda$*-module structure and, for each* $\mu \in M$, a right $B_\mu$*-module structure, all these module structures being compatible with one another.*

When the family $(B_\mu)$ (resp. $(A_\lambda)$) is empty, E is called a left (resp. right) multimodule. When Card(L) + Card(M) = 2, we say "*bimodule*" instead of "*multimodule*"; it is then often convenient to consider (as can always be done by replacing a ring of operators by its opposite, cf. no. 1) a bimodule as having a left module structure with respect to a ring A and a right module structure with respect to a ring B, the permutability of the laws then being expressed by the relation

It is then also said that E is an (A, B)-*bimodule*.

Two multimodule structures on a set E are said to be compatible if all the module structures on E which define one or the other of these multimodule structures are compatible with one another.