Bimodules

A bimodule over rings R and S, or an R–S-bimodule, is an abelian group M equipped with a left R-action and a right S-action that are compatible: for all r in R, m in M, s in S, (r·m)·s = r·(m·s). If R = S, this specializes to an R–R-bimodule. When R and S are algebras over a commutative ring k, M is a k-vector space with commuting left R-action and right S-action.

A standard way to view bimodules is through the tensor product. If M is an R–S-bimodule and N is an S–T-bimodule, their tensor product M ⊗_S N inherits an R–T-bimodule structure. This construction is associative up to natural isomorphism and provides a way to compose bimodules. The category of R–S-bimodules can be used to study functors between module categories: tensoring with a fixed bimodule M yields a functor from the category of right S-modules to the category of right R-modules.

Examples include the regular bimodule R itself, viewed with left and right multiplications by R, and more generally any bimodule arising from module structures that commute on both sides. Equivalently, an R–S-bimodule M can be regarded as a left module over the tensor product R ⊗ S^op.

Morita theory uses bimodules to relate module categories of different rings. An R–S-bimodule M that is finitely generated and projective on both sides and satisfies suitable reciprocity conditions can induce an equivalence between the categories of modules over R and S, showing that R and S have the same representation theory in a precise sense.

Bimodules play a central role in representation theory, algebraic topology, and category theory as a natural framework for morphisms between different algebraic structures.