Ring Multimodule

Let $R_{i}$ be an $I$-indexed family of Rings. A left $R_{i}$ multimodule is an Abelian Group $M$ that is an $R_{i}$ module for each $i:I$, such that all the distinct actions commute with one another:

• $\forall(r_{i}:R_{i})(r_{j}:R_{j}).\;i\neq j\Rightarrow r_{i}\cdot(r_{j}\cdot x )=r_{j}\cdot(r_{i}\cdot x)$

Despite the name, multimodules are unrelated to Multilinear Maps, which are concerned with the product of multiple modules.

Examples

• A $(R,S)$ Bimodule is a multimodule over the family $(R,S^{mathrm{op}})$.