Bimodule

Template:Short description 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.

Definition

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

1. M is a left R-module with an operation · and a right S-module with an operation <math>*</math>.
2. For all r in R, s in S and m in M: <math display="block">(r\cdot m)*s = r\cdot (m*s) .</math>

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

Examples

• For positive integers n and m, the set M_n,m(R) of Template:Nowrap matrices of real numbers is an Template:Nowrap, where R is the ring M_n(R) of Template:Nowrap matrices, and S is the ring M_m(R) of Template:Nowrap matrices. Addition and multiplication are carried out using the usual rules of matrix addition and matrix multiplication; the heights and widths of the matrices have been chosen so that multiplication is defined. Note that M_n,m(R) itself is not a ring (unless Template:Nowrap), because multiplying an Template:Nowrap matrix by another Template:Nowrap matrix is not defined. The crucial bimodule property, that Template:Nowrap, is the statement that multiplication of matrices is associative (which, in the case of a matrix ring, corresponds to associativity).
• Any algebra A over a ring R has the natural structure of an R-bimodule, with left and right multiplication defined by Template:Nowrap and Template:Nowrap respectively, where Template:Nowrap is the canonical embedding of R into A.
• If R is a ring, then R itself can be considered to be an Template:Nowrap by taking the left and right actions to be multiplication – the actions commute by associativity. This can be extended to Rⁿ (the n-fold direct product of R).
• Any two-sided ideal of a ring R is an Template:Nowrap, with the ring multiplication both as the left and as the right multiplication.
• Any module over a commutative ring R has the natural structure of 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. (However, not all R-bimodules arise this way: other compatible right multiplications may exist.)
• If M is a left R-module, then M is an Template:Nowrap, where Z is the ring of integers. Similarly, right R-modules may be interpreted as Template:Nowrap. Any abelian group may be treated as a Template:Nowrap.
• If M is a right R-module, then the set Template:Nowrap of R-module endomorphisms is a ring with the multiplication given by composition. The endomorphism ring Template:Nowrap acts on M by left multiplication defined by Template:Nowrap. The bimodule property, that Template:Nowrap, restates that f is a R-module homomorphism from M to itself. Therefore any right R-module M is an Template:Nowrap-bimodule. Similarly any left R-module N is an Template:Nowrap-bimodule.
• If R is a subring of S, then S is an Template:Nowrap. It is also an Template:Nowrap and an Template:Nowrap.
• If M is an S-R-bimodule and N is an Template:Nowrap, then Template:Nowrap is an S-T-bimodule.

Further notions and facts

If M and N are R-S-bimodules, then a map Template:Nowrap 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 Template:Nowrap, where S^op is the opposite ring of S (where the multiplication is defined with the arguments exchanged). Bimodule homomorphisms are the same as homomorphisms of left Template:Nowrap 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 Template:Nowrap 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 Template:Nowrap and N is an Template:Nowrap, then the tensor product of M and N (taken over the ring S) is an Template:Nowrap 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 Template:Nowrap M and N are exactly bimodule homomorphisms, i.e. functions

<math>f: M \rightarrow N</math>

that satisfy

1. <math>f(m+m') = f(m)+ f(m')</math>
2. <math>f(r.m.s) = r.f(m).s</math>,

for Template:Nowrap, Template:Nowrap, and Template:Nowrap. One immediately verifies the interchange law for bimodule homomorphisms, i.e.

<math>(f'\otimes g')\circ (f\otimes g) = (f'\circ f)\otimes(g'\circ g) </math>

holds whenever either (and hence the other) side of the equation is defined, and where <math>\circ</math> is the usual composition of homomorphisms. In this interpretation, the category Template:Nowrap is exactly the monoidal category of Template:Nowrap 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 Template:Nowrap, which gives a monoidal embedding of the category Template:Nowrap into Template:Nowrap. The case that R is a field K is a motivating example of a symmetric monoidal category, in which case Template:Nowrap, the category of vector spaces over K, with the usual tensor product Template:Nowrap giving the monoidal structure, and with unit K. We also see that a monoid in Template:Nowrap is exactly an R-algebra.Template:Clarify<ref name=arXiv>Template:Cite arXiv</ref> Furthermore, if M is an Template:Nowrap and L is an Template:Nowrap, then the set Template:Nowrap of all S-module homomorphisms from M to L becomes a Template:Nowrap 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.

See also

• Profunctor

References

• Template:Cite book