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.
For example, the category of all R-S-bimodules is abelian, and the standard isomorphism theorems are valid for bimodules.
In particular, if R is a commutative ring, every left or right R-module is canonically an R-R-bimodule, which gives a monoidal embedding of the category R-Mod into Bimod(R, R).
[clarification needed][1] 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-bimodule in a natural fashion.
These statements extend to the derived functors Ext and Tor.