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.
Contents
Definition
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:
An R-R-bimodule is also known as an R-bimodule.
Examples
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
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. This is in fact a 2-category, in a canonical way—2 morphisms between R-S-bimodules M and N are exactly bimodule homomorphisms, i.e. functions
satisfying
-
f ( m + m ′ ) = f ( m ) + f ( m ′ ) -
f ( r m s ) = r f ( m ) s ,
for m ∈ M, r ∈ R, and s ∈ S. One immediately verifies the interchange law for bimodule homomorphisms, i.e.
holds whenever either (and hence the other) side of the equation is defined, and where ∘ is the usual composition of homomorphisms. In this interpretation, the category End(R) = Bimod(R, R) is exactly the monoidal category of R-R-bimodules with the usual tensor product over R the tensor product of the category. 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). The case that R is a field K is a motivating example of a symmetric monoidal category, in which case R-Mod = K-Vect, the category of vector spaces over K, with the usual tensor product
Profunctors can be seen as a categorical generalization of bimodules.
Note that bimodules are not at all related to bialgebras.