Bbimodule

Bbimodule is typically a misspelling of bimodule, a fundamental concept in algebra. A bimodule over rings (or algebras) is an abelian group that carries both a left action of one ring and a right action of another, with the two actions commuting.

Formally, if R and S are rings, an (R,S)-bimodule M consists of a left R-action and a right S-action on M such that (r m) s = r (m s) for all r in R, m in M, and s in S. When R = S, M is simply an R-bimodule. If M is unital, the actions satisfy 1_R m = m and m 1_S = m.

Common examples include: the ring R itself regarded as an (R,R)-bimodule via ring multiplication; an (R,S)-bimodule constructed from a ring homomorphism f: R → S, where r·m·s = f(r) m s; and subsets of linear maps between modules that inherit compatible left and right actions.

One important construction is the tensor product. If M is an (R,S)-bimodule and N is an (S,T)-bimodule, their tensor product M ⊗_S N becomes an (R,T)-bimodule. This operation underpins functors between module categories and plays a central role in Morita theory, which studies when different rings have equivalent categories of modules.

In addition, bimodules form a bicategory whose objects are rings, 1-morphisms are bimodules, and 2-morphisms are bimodule homomorphisms. They appear in representation theory, algebraic geometry, and the study of ring extensions and equivalences.

See also: bimodule, module, ring, Morita theory, tensor product, bicategory.