Bimodule
Encyclopedia
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.
, then an R-S-bimodule is an abelian group M such that:
An R-R-bimodule is also known as an R-bimodule.
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 theorem
s 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 functor
s Ext
and Tor
.
Profunctor
s can be seen as a categorical generalization of bimodules.
Note that bimodules are not at all related to bialgebra
s.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
a bimodule is an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
that is both a left and a right module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
, 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 ringsRing (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
, 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
- For positive integers n and m, the set Mn,m(R) of n × m matricesMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
of real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s is an R-S bimodule, where R is the ring Mn(R) of n × n matrices, and S is the ring Mm(R) of m × m matrices. Addition and multiplication are carried out using the usual rules of matrix additionMatrix additionIn mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum....
and matrix multiplicationMatrix multiplicationIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
; the heights and widths of the matrices have been chosen so that multiplication is defined. Note that Mn,m(R) itself is not a ring (unless n = m), because multiplying an n × m matrix by another n × m matrix is not defined. The crucial bimodule property, that (r x)s = r(x s), is the statement that multiplication of matrices is associative. - If R is a ring, then R itself is an R-bimodule, and so is Rn (the n-fold direct productProduct of ringsIn mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product Πi in I Ri can be turned into a ring by defining the operations coordinatewise, i.e...
of R). - Any two-sided idealIdeal (ring theory)In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
of a ring R is an R-bimodule. - Any module over a commutative ringCommutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
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. (However, not all R-bimodules arise this way.) - If M is a left R-module, then M is an R-Z bimodule, where Z is the ring of integerIntegerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s. Similarly, right R-modules may be interpreted as Z-R bimodules, and indeed an abelian group may be treated as a Z-Z bimodule. - If R is a subringSubringIn mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
of S, then S is an R-bimodule. It is also an R-S and an 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
, 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 categoryCategory theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
of all R-S bimodules is abelian
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
, and the standard isomorphism theorem
Isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures...
s 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
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
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 functor
Derived functor
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...
s Ext
Ext functor
In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...
and Tor
Tor functor
In homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology....
.
Profunctor
Profunctor
In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules. They are related to the notion of correspondences.- Definition :...
s can be seen as a categorical generalization of bimodules.
Note that bimodules are not at all related to bialgebra
Bialgebra
In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a coalgebra, such that these structures are compatible....
s.