Bimodule - Further Notions and Facts

Further Notions and Facts

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, where Sop is the opposite ring of S (with the multiplication turned around). Bimodule homomorphisms are the same as homomorphisms of left 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.

There are however some new effects in the world of bimodules, especially when it comes to the tensor product: if M is an R-S bimodule and N is an S-T bimodule, then the tensor product of M and N (taken over the ring S) is an R-T bimodule in a natural fashion. This tensor product of bimodules is associative (up to a unique canonical isomorphism), and one can hence construct a category whose objects are the rings and whose morphisms are the bimodules. Furthermore, if M is an R-S bimodule and L is an T-S bimodule, then the set HomS(M,L) of all S-module homomorphisms from M to L becomes a T-R module in a natural fashion. These statements extend to the derived functors Ext and Tor.

Profunctors can be seen as a categorical generalization of bimodules.

Note that bimodules are not at all related to bialgebras.

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