bimodule

A bimodule over rings R and S is a abelian group M endowed with a left action of R and a right action of S that are compatible: for all r in R, m in M, s in S, we have r·(m·s) = (r·m)·s. If R = S, an (R,S)-bimodule is called an R-R bimodule. When the rings have identities and M is unital, 1_R·m = m = m·1_S for all m in M.

Examples include the ring R itself, viewed as an R-R-bimodule with the actions given by left and right multiplication, r·m = rm and m·s = ms. More generally, any bimodule M in which the left and right actions commute (r·(m·s) = (r·m)·s) provides an (R,S)-bimodule. If M is an (R,S)-bimodule and N is an (S,T)-bimodule, then the tensor product M ⊗_S N inherits an (R,T)-bimodule structure, illustrating how bimodules compose.

Applications and importance: bimodules serve as a bridge between module categories. If M is an (R,S)-bimodule, then the tensor product functor − ⊗_S M maps right S-modules to left R-modules, providing a way to transfer structures across rings. Bimodules are central in Morita theory: rings R and S are Morita equivalent if their module categories are equivalent, typically realized by a suitable pair of bimodules that induce equivalences of categories.

See also: Morita theory, tensor product of modules, module category, endomorphism rings.