ABbimodule

ABbimodule refers to a mathematical structure that is simultaneously a module over two rings, typically denoted A and B. The term “bimodule” indicates that the object carries both a left module structure with respect to one ring and a right module structure with respect to another. In an ABbimodule M, the left action of A and the right action of B must satisfy a compatibility condition: for all elements a in A, b in B, and m in M, the equation (a·m)·b = a·(m·b) holds. This condition ensures that the two module actions interact coherently.

The concept of a bimodule generalizes the notion of a module by allowing the scalar multiplication to come from two different rings, potentially with different characteristics. Bimodules arise naturally in many areas of algebra and representation theory. For example, if A and B are algebras over a commutative ring k, then an ABbimodule can be viewed as a k-vector space equipped with compatible left and right algebra actions. When A equals B, an ABbimodule is simply an A-bimodule, which can be used to study extensions and tensor products of algebras.

Common examples of ABbimodules include the ring A itself regarded as an (A, A)-bimodule with multiplication on both sides, or the set of linear maps from one module to another when the source and target carry module structures over different rings. Bimodules also play a role in Morita theory, which investigates when two rings have equivalent module categories by means of an invertible bimodule that implements a categorical equivalence. In homological algebra, bimodules serve as coefficient modules in Hochschild cohomology, providing a framework for studying derivations and deformations of associative algebras.