Bimodule

In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules.

If R and S are two rings, then an R-S-bimodule is an abelian group M such that:

An R-R-bimodule is also known as an R-bimodule.

If M and N are R-S-bimodules, then a map f : M → N is a bimodule homomorphism if it is both a homomorphism of left R-modules and of right S-modules.

An R-S-bimodule is actually the same thing as a left module over the ring ${\displaystyle R\otimes _{\mathbb {Z} }S^{op}}$, where S^op is the opposite ring of S (with the multiplication turned around). Bimodule homomorphisms are the same as homomorphisms of left ${\displaystyle R\otimes _{\mathbb {Z} }S^{op}}$ modules. Using these facts, many definitions and statements about modules can be immediately translated into definitions and statements about bimodules. For example, the category of all R-S-bimodules is abelian, and the standard isomorphism theorems are valid for bimodules.

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