Bimodule

Summary

A bimodule over two rings $R$ and $S$ is a left module over $R$ that is also a right module over $S$. The two actions must commute. Bimodules are the natural objects between two rings: tensoring with an $(R, S)$-bimodule turns right $R$-modules into right $S$-modules. The two-sided ideals of a ring are exactly its sub-bimodules. Bimodules underlie Morita theory and Hochschild cohomology.