Bimodule
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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.
Formally, if R and S are two rings, then an R-S-bimodule is an abelian group M such that:
- M is a left R-module and a right S-module.
- For all r in R, s in S and m in M:
- (rm)s = r(ms).
An R-R-bimodule is also known as an R-bimodule.
Examples
- R is a R-bimodule, and so is Rn.
- A two-sided ideal of R is an R-bimodule.
- Any module over a commutative ring R is automatically a bimodule. For example, if M is a left module, we can define multiplication on the right to be the same as multiplication on the left. (Note that not all R-bimodules arise this way.)
- If M is a left R module, then M is an R-Z bimodule.
- If R is a subring of S, then S is an R-bimodule. (It is also an R-S and S-R bimodule.)
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 R×Sop, where Sop is the opposite ring of S (with the multiplication turned around). Bimodule homomorphisms are the same as homomorphisms of left R×Sop 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.
Note that bimodules are not at all related to bialgebras.