In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, 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
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 such that:
# ''M'' is a left ''R''-module with an operation · and a right ''S''-module with an operation
.
# 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
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s ''n'' and ''m'', the set ''M''
''n'',''m''(R) of
matrices of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s is an , where ''R'' is the ring ''M''
''n''(R) of matrices, and ''S'' is the ring ''M''
''m''(R) of 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 ), because multiplying an matrix by another matrix is not defined. The crucial bimodule property, that , 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 and respectively, where is the canonical embedding of ''R'' into ''A''.
* If ''R'' is a ring, then ''R'' itself can be considered to be an by taking the left and right actions to be multiplication – the actions commute by associativity. This can be extended to ''R''
''n'' (the ''n''-fold
direct product of ''R'').
* Any two-sided
ideal of a ring ''R'' is an , with the ring multiplication both as the left and as the right multiplication.
* Any module over a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
''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 , where Z is the
ring of integers. Similarly, right ''R''-modules may be interpreted as . Any abelian group may be treated as a .
* If ''M'' is a right ''R''-module, then the set of ''R''-module
endomorphisms is a ring with the multiplication given by composition. The endomorphism ring acts on ''M'' by left multiplication defined by . The bimodule property, that , restates that ''f'' is a ''R''-module homomorphism from ''M'' to itself. Therefore any right ''R''-module ''M'' is an -bimodule. Similarly any left ''R''-module ''N'' is an -bimodule.
* If ''R'' is a
subring of ''S'', then ''S'' is an . It is also an and an .
* If ''M'' is an ''S''-''R''-bimodule and ''N'' is an , then is an ''S''-''T''-bimodule.
Further notions and facts
If ''M'' and ''N'' are ''R''-''S''-bimodules, then a map 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 ''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 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 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
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
: if ''M'' is an and ''N'' is an , then the tensor product of ''M'' and ''N'' (taken over the ring ''S'') is an in a natural fashion. This tensor product of bimodules is
associative (
up to a unique canonical
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
), 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 ''M'' and ''N'' are exactly bimodule homomorphisms, i.e. functions
:
that satisfy
#
#
,
for , , and . 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 is exactly the
monoidal category
In mathematics, a monoidal category (or tensor category) is a category (mathematics), category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an Object (cate ...
of with the usual
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
over ''R'' the tensor product of the category. In particular, if ''R'' is a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
, every left or right ''R''-module is canonically an , which gives a monoidal embedding of the category into . The case that ''R'' is a
field ''K'' is a motivating example of a symmetric monoidal category, in which case , the
category of vector spaces over ''K'', with the usual tensor product giving the monoidal structure, and with unit ''K''. We also see that a
monoid in is exactly an ''R''-algebra.
Furthermore, if ''M'' is an and ''L'' is an , then the
set of all ''S''-module homomorphisms from ''M'' to ''L'' becomes a 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
bialgebra
In mathematics, a bialgebra over a Field (mathematics), field ''K'' is a vector space over ''K'' which is both a unital algebra, unital associative algebra and a coalgebra, counital coassociative coalgebra. The algebraic and coalgebraic structure ...
s.
See also
*
Profunctor
References
* {{cite book , author=Jacobson, N. , author-link=Nathan Jacobson, title=Basic Algebra II , publisher=W. H. Freeman and Company , year=1989 , pages=133–136 , isbn=0-7167-1933-9
Module theory