Editing Bimodule

{{Short description|Abelian group equipped with compatible ring action on both sides}}
In [[abstract algebra]], a '''bimodule''' is an [[abelian group]] that is both a left and a right [[module (mathematics)|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 [[ring (mathematics)|ring]]s, then an ''R''-''S''-'''bimodule''' is an abelian group {{nowrap|(''M'', +)}} such that:
# ''M'' is a left ''R''-module with an operation '''·''' and a right ''S''-module with an operation <math>*</math>.
#  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 [[integer]]s ''n'' and ''m'', the set ''M''<sub>''n'',''m''</sub>('''R''') of {{nowrap|''n'' × ''m''}} [[matrix (mathematics)|matrices]] of [[real number]]s is an {{nowrap|''R''-''S''-bimodule}}, where ''R'' is the ring ''M''<sub>''n''</sub>('''R''') of {{nowrap|''n'' × ''n''}} matrices, and ''S'' is the ring ''M''<sub>''m''</sub>('''R''') of {{nowrap|''m'' × ''m''}} 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''<sub>''n'',''m''</sub>('''R''') itself is not a ring (unless {{nowrap|1=''n'' = ''m''}}), because multiplying an {{nowrap|''n'' × ''m''}} matrix by another {{nowrap|''n'' × ''m''}} matrix is not defined. The crucial bimodule property, that {{nowrap|1=(''r''.''x'').''s'' = ''r''.(''x''.''s'')}}, 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 {{nowrap|1=''r''.''a'' = ''φ''(''r'')''a''}} and {{nowrap|1=''a''.''r'' = ''aφ''(''r'')}} respectively, where {{nowrap|''φ'' : ''R'' → ''A''}} is the canonical embedding of ''R'' into ''A''.
* If ''R'' is a ring, then ''R'' itself can be considered to be an {{nowrap|''R''-''R''-bimodule}} by taking the left and right actions to be multiplication – the actions commute by associativity. This can be extended to ''R''<sup>''n''</sup> (the ''n''-fold [[product of rings|direct product]] of ''R'').
* Any two-sided [[ideal (ring theory)|ideal]] of a ring ''R'' is an {{nowrap|''R''-''R''-bimodule}}, 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 {{nowrap|''R''-'''Z'''-bimodule}}, where '''Z''' is the [[Integer#Algebraic properties|ring of integers]]. Similarly, right ''R''-modules may be interpreted as {{nowrap|'''Z'''-''R''-bimodules}}. Any abelian group may be treated as a {{nowrap|'''Z'''-'''Z'''-bimodule}}.
* If ''M'' is a right ''R''-module, then the set {{nowrap|End<sub>''R''</sub>(''M'')}} of ''R''-module [[endomorphism]]s is a ring with the multiplication given by composition. The endomorphism ring {{nowrap|End<sub>''R''</sub>(''M'')}}  acts on ''M'' by left multiplication defined by {{nowrap|1=''f''.''x'' = ''f''(''x'')}}. The bimodule property, that {{nowrap|1=(''f''.''x'').''r'' = ''f''.(''x''.''r'')}}, restates that ''f'' is  a ''R''-module homomorphism from ''M'' to itself. Therefore any right ''R''-module ''M'' is an {{nowrap|End<sub>''R''</sub>(''M'')-''R''}}-bimodule. Similarly any left ''R''-module ''N'' is an {{nowrap|''R''-End<sub>''R''</sub>(''N'')<sup>op</sup>}}-bimodule.
* If ''R'' is a [[subring]] of ''S'', then ''S'' is an {{nowrap|''R''-''R''-bimodule}}. It is also an {{nowrap|''R''-''S''-}} and an {{nowrap|''S''-''R''-bimodule}}.
* If ''M'' is an ''S''-''R''-bimodule and ''N'' is an {{nowrap|''R''-''T''-bimodule}}, then {{nowrap|''M'' ⊗<sub>''R''</sub> ''N''}} is an ''S''-''T''-bimodule.

== Further notions and facts ==
If ''M'' and ''N'' are ''R''-''S''-bimodules, then a map {{nowrap|''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 {{nowrap|''R'' ⊗<sub>'''Z'''</sub> ''S''<sup>op</sup>}}, where ''S''<sup>op</sup> is the [[opposite ring]] of ''S'' (where the multiplication is defined with the arguments exchanged). Bimodule homomorphisms are the same as homomorphisms of left {{nowrap|''R'' ⊗<sub>'''Z'''</sub> ''S''<sup>op</sup>}} modules. Using these facts, many definitions and statements about modules can be immediately translated into definitions and statements about bimodules. For example, the [[category (mathematics)|category]] of all {{nowrap|''R''-''S''-bimodules}} is [[abelian category|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 of modules|tensor product]]: if ''M'' is an {{nowrap|''R''-''S''-bimodule}} and ''N'' is an {{nowrap|''S''-''T''-bimodule}}, then the tensor product of ''M'' and ''N'' (taken over the ring ''S'') is an {{nowrap|''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 {{nowrap|''R''-''S''-bimodules}} ''M'' and ''N'' are exactly bimodule homomorphisms, i.e. functions
: <math>f: M \rightarrow N</math>
that satisfy
# <math>f(m+m') = f(m)+ f(m')</math>  
# <math>f(r.m.s) = r.f(m).s</math>,
for {{nowrap|''m'' ∈ ''M''}}, {{nowrap|''r'' ∈ ''R''}}, and {{nowrap|''s'' ∈ ''S''}}. 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 {{nowrap|1='''End'''(''R'') = '''Bimod'''(''R'', ''R'')}} is exactly the [[monoidal category]] of {{nowrap|''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 {{nowrap|''R''-''R''-bimodule}}, which gives a monoidal embedding of the category {{nowrap|1= ''R''-'''Mod'''}} into {{nowrap|1='''Bimod'''(''R'', ''R'')}}. The case that ''R'' is a [[field (mathematics)|field]] ''K'' is a motivating example of a symmetric monoidal category, in which case {{nowrap|1=''R''-'''Mod''' = ''K''-'''Vect'''}}, the [[category of vector spaces]] over ''K'', with the usual tensor product {{nowrap|1=⊗ = ⊗<sub>''K''</sub>}} giving the monoidal structure, and with unit ''K''. We also see that a [[monoid (category theory)|monoid]] in {{nowrap|'''Bimod'''(''R'', ''R'')}} is exactly an ''R''-algebra.{{clarify|reason=Are we still requiring that ''R'' is commutative?|date=June 2024}}<ref name=arXiv>{{cite arXiv|last1=Street|first1=Ross|title=Categorical and combinatorial aspects of descent theory|date=20 Mar 2003|eprint=math/0303175}}</ref>
Furthermore, if ''M'' is an {{nowrap|''R''-''S''-bimodule}} and ''L'' is an {{nowrap|''T''-''S''-bimodule}}, then the [[set (mathematics)|set]] {{nowrap|Hom<sub>''S''</sub>(''M'', ''L'')}} of all ''S''-module homomorphisms from ''M'' to ''L'' becomes a {{nowrap|''T''-''R''-bimodule}} in a natural fashion. These statements extend to the [[derived functor]]s [[Ext functor|Ext]] and [[Tor functor|Tor]].

[[Profunctor]]s can be seen as a categorical generalization of bimodules.

Note that bimodules are not at all related to [[bialgebra]]s.

== See also ==
* [[Profunctor]]

== References ==
{{reflist}}
* {{cite book | author=Jacobson, N. | author-link=Nathan Jacobson| title=Basic Algebra II | publisher=W. H. Freeman and Company | year=1989 | pages=133&ndash;136 | isbn=0-7167-1933-9 }}
[[Category:Module theory]]