Abimodule

An A-bimodule is a module that is equipped with both a left and a right action of a ring A, such that the two actions commute. Concretely, M is an abelian group with actions A × M → M and M × A → M satisfying (a m) b = a (m b) for all a, b in A and m in M, and m 1 = m if A has a multiplicative identity. Equivalently, M is a module over the enveloping algebra A^e = A ⊗ Z A^op.

A-bimodules arise naturally in several contexts. The category of A-bimodules is the same as the category of modules over the enveloping algebra A^e. The most basic example is A itself, viewed as a bimodule with the regular left and right multiplication. More generally, any two-sided ideal I of A becomes an A-bimodule under the induced left and right multiplications. Quotients such as A/I (for a two-sided ideal I) also carry natural A-bimodule structures. If A is commutative, every A-bimodule is simply an A-module with a single action, since left and right actions coincide.

Morphisms of A-bimodules are maps that preserve both the left and right actions: f(a m b) = a f(m) b for all a, b in A and m in M. The notion is central in many areas of algebra; bimodules are the natural coefficients for Hochschild cohomology, appear in Morita theory, and underpin various constructions in representation theory and deformation theory.