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| In [[mathematics]], the '''tensor product of modules''' is a construction that allows arguments about [[bilinear map|bilinear]] maps (e.g. multiplication) to be carried out in terms of linear maps ([[module homomorphism]]s). The module construction is analogous to the construction of the [[tensor product]] of [[vector space]]s, but can be carried out for a pair of [[module (mathematics)|modules]] over a [[commutative ring]] resulting in a third module, and also for a pair of a left-module and a right-module over any [[ring (mathematics)|ring]], with result an [[abelian group]]. Tensor products are important in areas of [[abstract algebra]], [[homological algebra]], [[algebraic topology]] and [[algebraic geometry]]. The [[universal property]] of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via [[linear operator|linear operations]]. The tensor product of an algebra and a module can be used for [[extension of scalars]]. For a commutative ring, the tensor product of modules can be iterated to form the [[tensor algebra]] of a module, allowing one to define multiplication in the module in a universal way.
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| ==Multilinear mappings==
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| For a ring ''R'', a right ''R''-module ''M<sub>R</sub>'', a left ''R''-module ''<sub>R</sub>N'', and an abelian group ''Z'', a '''bilinear map''' or '''balanced product''' from {{nowrap|''M'' × ''N''}} to ''Z'' is a function {{nowrap|''φ'': ''M'' × ''N'' → ''Z''}} such that for all ''m'', ''m''′ in ''M'', ''n'', ''n''′ in ''N'', and ''r'' in ''R'':
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| # ''φ''(''m'' + ''m''′, ''n'') = ''φ''(''m'', ''n'') + ''φ''(''m''′, ''n'')
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| # ''φ''(''m'', ''n'' + ''n''′) = ''φ''(''m'', ''n'') + ''φ''(''m'', ''n''′)
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| # ''φ''(''m'' · ''r'', ''n'') = ''φ''(''m'', ''r'' · ''n'')
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| The set of all such bilinear maps from {{nowrap|''M'' × ''N''}} to ''Z'' is denoted by {{nowrap|Bilin(''M'', ''N''; ''Z'')}}.
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| Property 3 differs slightly from the definition for vector spaces. This is necessary because ''Z'' is only assumed to be an abelian group, so {{nowrap|''r'' · ''φ''(''m'', ''n'')}} would not make sense.
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| If ''φ'', ''ψ'' are bilinear maps, then {{nowrap|''φ'' + ''ψ''}} is a bilinear map, and −''φ'' is a bilinear map, when these operations are defined [[pointwise]]. This turns the set {{nowrap|Bilin(''M'', ''N''; ''Z'')}} into an abelian group. The neutral element is the zero mapping.
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| For ''M'' and ''N'' fixed, the map {{nowrap|''Z'' ↦ Bilin(''M'', ''N''; ''Z'')}} is a [[functor]] from the [[category of abelian groups]] to the [[category of sets]]. The morphism part is given by mapping a group homomorphism {{nowrap|''g'' : ''Z'' → ''Z''′}} to the function {{nowrap|''φ'' ↦ ''g'' ∘ ''φ''}}, which goes from {{nowrap|Bilin(''M'', ''N''; ''Z'')}} to {{nowrap|Bilin(''M'', ''N''; ''Z''′)}}.
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| ==Definition==
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| Let ''M'',''N'' and ''R'' be as in the previous section. The '''tensor product''' over ''R''
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| :<math>M \otimes_R N</math>
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| is an [[abelian group]] together with a bilinear map (in the sense defined above)
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| :<math>\otimes : M \times N \to M \otimes_{R} N</math>
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| which is [[universal property|universal]] in the following sense:<ref>Hazewinkel, et al. (2004), [http://books.google.com.br/books?id=AibpdVNkFDYC&pg=PA95 p. 95], Prop. 4.5.1.</ref>
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| [[File:Tensor product of modules.png|200px|right]]
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| :For every abelian group ''Z'' and every bilinear map
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| ::<math>f: M \times N \to Z\,</math>
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| :there is a '''unique''' group homomorphism
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| ::<math> \tilde{f}: M \otimes_R N \to Z</math>
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| :such that
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| ::<math>\tilde{f} \circ \otimes = f.</math>
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| As with all [[Universal property#Existence and uniqueness|universal properties]], the above property defines the tensor product uniquely [[up to]] a unique isomorphism: any other object and bilinear map with the same properties will be isomorphic to {{nowrap|''M'' ⊗<sub>''R''</sub> ''N''}} and ⊗. The definition does not prove the existence of {{nowrap|''M'' ⊗<sub>''R''</sub> ''N''}}; see below for a construction.
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| The tensor product can also be defined as a [[representable functor|representing object]] for the functor {{nowrap|''Z'' → Bilin<sub>''R''</sub>(''M'',''N'';''Z'')}}. This is equivalent to the universal mapping property given above.
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| Strictly speaking, the ring used to form the tensor should be indicated: most modules can be considered as modules over several different rings or over the same ring with a different actions of the ring on the module elements. For example, it can be shown that {{nowrap|'''R''' ⊗<sub>'''R'''</sub> '''R'''}} and {{nowrap|'''R''' ⊗<sub>'''Z'''</sub> '''R'''}} are completely different from each other. However in practice, whenever the ring is clear from context, the subscript denoting the ring may be dropped.
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| ==Examples==
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| Consider the [[rational number]]s '''Q''' and the [[modular arithmetic|integers modulo ''n'']] '''Z'''<sub>''n''</sub>. As with any abelian group, both can be considered as modules over the [[integer]]s, '''Z'''.
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| Let ''B'': '''Q''' × '''Z'''<sub>''n''</sub> → ''M'' be a '''Z'''-bilinear operator. Then ''B''(''q'', ''k'') = ''B''(''q''/''n'', ''nk'') = ''B''(''q''/''n'', 0) = 0, so every bilinear operator is identically zero. Therefore, if we define <math>{\mathbf Q} \otimes {\mathbf Z}_n</math> to be the trivial module, and <math>\tilde{f}</math> to be the zero bilinear function, then we see that the properties for the tensor product are satisfied. Therefore, the tensor product of '''Q''' and '''Z'''<sub>''n''</sub> is {0}.<ref>Hazewinkel, et al. (2004), [http://books.google.com.br/books?id=AibpdVNkFDYC&pg=PA97 p. 97], Ex. 4.5.1.</ref>
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| An [[abelian group]] is a '''Z'''-module, which allows the theory of abelian groups to be subsumed in that of modules.<ref>{{cite book |first=Nathan |last=Jacobson |title=Basic Algebra |volume=I |edition=2nd |publisher=Dover |year=2009 |page=164 }}</ref> The tensor product of '''Z'''-modules is sometimes termed the '''tensor product of abelian groups'''.
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| ==Construction==
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| The construction of ''M'' ⊗ ''N'' takes a quotient of a [[free abelian group]] with basis the symbols ''m'' ⊗ ''n'' for ''m'' in ''M'' and ''n'' in ''N'' by the subgroup generated by all elements of the form
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| # −(''m''+''m′'') ⊗ ''n'' + ''m'' ⊗ ''n'' + ''m′'' ⊗ ''n''
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| # −''m'' ⊗ (''n''+''n′'') + ''m'' ⊗ ''n'' + ''m'' ⊗ ''n′''
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| # (''m''·''r'') ⊗ ''n'' − ''m'' ⊗ (''r''·''n'')
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| where ''m'',''m′'' in ''M'', ''n'',''n′'' in ''N'', and ''r'' in ''R''. The function which takes (''m'',''n'') to the coset containing ''m'' ⊗ ''n'' is bilinear, and the subgroup has been chosen minimally so that this map is bilinear.
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| The [[direct product]] of ''M'' and ''N'' is rarely isomorphic to the tensor product of ''M'' and ''N''. When ''R'' is not commutative, then the tensor product requires that ''M'' and ''N'' be modules on opposite sides, while the direct product requires they be modules on the same side. In all cases the only function from ''M'' × ''N'' to ''Z'' which is both linear and bilinear is the zero map.
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| ==Relationship to flat modules==
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| In general, <math>-\otimes_R-:\mathrm{Mod}\mbox{--}R\times R\mbox{--}\mathrm{Mod}\rightarrow \mathrm{Ab}</math> is a [[bifunctor]] which accepts a right and a left ''R'' module pair as input, and assigns them to the tensor product in the [[category of abelian groups]].
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| By fixing a right ''R'' module ''M'', a functor <math>M\otimes_R-:R\mbox{--}\mathrm{Mod}\rightarrow \mathrm{Ab}</math> arises, and symmetrically a left ''R'' module ''N'' could be fixed to create a functor <math>-\otimes_RN:\mathrm{Mod}\mbox{--}R\rightarrow \mathrm{Ab}</math>. Unlike the [[Hom bifunctor]] <math>\mathrm{Hom}_R(-,-)</math>, the tensor functor is [[covariant functor|covariant]] in both inputs.
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| It can be shown that ''M''⊗- and -⊗''N'' are always [[right exact functor]]s, but not necessarily left exact. By definition, a module ''T'' is a [[flat module]] if ''T''⊗- is an exact functor.
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| If {''m''<sub>''i''</sub>}<sub>''i''∈''I''</sub> and {''n''<sub>''j''</sub>}<sub>''j''∈''J''</sub> are generating sets for ''M'' and ''N'', respectively, then {''m''<sub>''i''</sub>⊗''n''<sub>''j''</sub>}<sub>''i''∈''I'',''j''∈''J''</sub> will be a generating set for ''M''⊗''N''. Because the tensor functor ''M''⊗<sub>''R''</sub>- sometimes fails to be left exact, this may not be a minimal generating set, even if the original generating sets are minimal.
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| When the tensor products are taken over a field ''F'' so that -⊗- is exact in both positions, and the generating sets are bases of ''M'' and ''N'', it is true that <math> \{m_i \otimes n_j \mid i\in I, j \in J\}</math> indeed forms a basis for ''M''⊗<sub>''F''</sub> ''N''.
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| ==Several modules==
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| It is possible to generalize the definition to a tensor product of any number of spaces. For example, the universal property of
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| :''M''<sub>1</sub> ⊗ ''M''<sub>2</sub> ⊗ ''M''<sub>3</sub>
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| is that each trilinear map on
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| :''M''<sub>1</sub> × ''M''<sub>2</sub> × ''M''<sub>3</sub> → ''Z''
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| corresponds to a unique linear map
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| :''M''<sub>1</sub> ⊗ ''M''<sub>2</sub> ⊗ ''M''<sub>3</sub> → ''Z''.
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| The binary tensor product is associative: (''M''<sub>1</sub> ⊗ ''M''<sub>2</sub>) ⊗ ''M''<sub>3</sub> is naturally isomorphic to ''M''<sub>1</sub> ⊗ (''M''<sub>2</sub> ⊗ ''M''<sub>3</sub>). The tensor product of three modules defined by the universal property of trilinear maps is isomorphic to both of these iterated tensor products.
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| ==Additional structure== | |
| The tensor product, as defined, is an abelian group, but in general, it does not immediately have an ''R''-module structure. However, if ''M'' is an (''S'',''R'')-[[bimodule]], then ''M''⊗<sub>''R''</sub>''N'' can be made into a left ''S''-module using the obvious operation ''s''(''m''⊗''n'')=(''sm''⊗''n''). Similarly, if ''N'' is an (''R'',''T'')-bimodule, then ''M''⊗<sub>''R''</sub>''N'' is a right ''T''-module using the operation (''m''⊗''n'')''t''=(''m''⊗''nt''). If ''M'' and ''N'' each have bimodule structures as above, then ''M''⊗<sub>''R''</sub>''N'' is an (''S'',''T'')-bimodule. In the case where ''R'' is a commutative ring, all of its modules can be thought of as (''R'',''R'')-bimodules, and then ''M''⊗<sub>''R''</sub>''N'' can be made into an ''R''-module as described. In the construction of the tensor product over a commutative ring ''R'', the multiplication operation can either be defined ''a posteriori'' as just described, or can be built in from the start by forming the quotient of a free ''R''-module by the submodule generated by the elements given above for the general construction, augmented by the elements r (''m'' ⊗ ''n'') − ''m'' ⊗ (''r''·''n''), or equivalently the elements (''m''·''r'') ⊗ ''n'' − r (''m'' ⊗ ''n'').
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| If {''m''<sub>''i''</sub>}<sub>''i''∈''I''</sub> and {''n''<sub>''j''</sub>}<sub>''j''∈''J''</sub> are generating sets for ''M'' and ''N'', respectively, then {''m''<sub>''i''</sub>⊗''n''<sub>''j''</sub>}<sub>''i''∈''I'',''j''∈''J''</sub> will be a generating set for ''M''⊗''N''. Because the tensor functor ''M''⊗<sub>''R''</sub>- is right [[exact functor|exact]], but sometimes not left exact, this may not be a minimal generating set, even if the original generating sets are minimal. If ''M'' is a [[flat module]], the functor <math>M\otimes_R-</math> is exact by the very definition of a flat module. If the tensor products are taken over a field ''F'', we are in the case of vector spaces as above. Since all ''F'' modules are flat, the [[bifunctor]]<math>-\otimes_R-</math> is exact in both positions, and the two given generating sets are bases, then
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| <math> \{m_i \otimes n_j \mid i\in I, j \in J\}</math> indeed forms a basis for ''M'' ⊗<sub>''F''</sub> ''N''.
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| If ''S'' and ''T'' are commutative ''R''-algebras, then ''S'' ⊗<sub>R</sub> ''T'' will be a commutative ''R''-algebra as well, with the multiplication map defined by (''m''<sub>1</sub> ⊗ ''m''<sub>2</sub>) (''n''<sub>1</sub> ⊗ ''n''<sub>2</sub>) = (''m''<sub>1</sub>''n''<sub>1</sub> ⊗ ''m''<sub>2</sub>''n''<sub>2</sub>) and extended by linearity. In this setting, the tensor product become a [[fibered coproduct]] in the category of ''R''-algebras. Note that any ring is a ''Z''-algebra, so we may always take ''M'' ⊗<sub>'''Z'''</sub> ''N''.
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| If ''<sub>S<sub>1</sub></sub>M<sub>R</sub>'' is an ''S<sub>1</sub>-R''-bimodule, then there is a unique left ''S<sub>1</sub>''-module structure on ''M''⊗''N'' which is compatible with the tensor map ⊗:''M''×''N''→''M''⊗<sub>R</sub>''N''. Similarly, if ''<sub>R</sub>N<sub>S<sub>2</sub></sub>'' is an ''R-S<sub>2</sub>''-bimodule, then there is a unique right ''S<sub>2</sub>''-module structure on ''M''⊗<sub>R</sub>''N'' which is compatible with the tensor map.{{Citation needed|date=February 2011}}
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| If ''M'' and ''N'' are both ''R''-modules over a commutative ring, then their tensor product is again an ''R''-module. If ''R'' is a ring, ''<sub>R</sub>M'' is a left ''R''-module, and the [[commutator]]
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| :''rs'' − ''sr''
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| of any two elements ''r'' and ''s'' of ''R'' is in the [[Annihilator (ring theory)|annihilator]] of ''M'', then we can make ''M'' into a right ''R'' module by setting
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| :''mr'' = ''rm''.
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| The action of ''R'' on ''M'' factors through an action of a quotient commutative ring. In this case the tensor product of ''M'' with itself over ''R'' is again an ''R''-module. This is a very common technique in commutative algebra.
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| ==See also==
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| *[[Tor functor]]
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| *[[Tensor product of algebras]]
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| *[[Tensor product of fields]]
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| == Notes ==
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| <references/>
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| == References ==
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| {{refimprove|date=February 2008}}<!-- no inline cites, and this reference was given as "Further reading" as if this was a textbook. -->
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| * {{citation|first1=D.G.|last1=Northcott|authorlink1=Douglas Northcott|title=Multilinear Algebra|publisher=Cambridge University Press|year=1984|isbn=613-0-04808-4}}.
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| * {{citation|first1=Michiel|last1=Hazewinkel|authorlink1=Michiel Hazewinkel|first2=Nadezhda Mikhaĭlovna|last2=Gubareni|authorlink2=Nadezhda Mikhaĭlovna|first3=Nadiya|last3=Gubareni|authorlink3=Nadiya Gubareni|first4=Vladimir V.|last4=Kirichenko|authorlink4=Vladimir V. Kirichenko|title=Algebras, rings and modules|publisher=Springer|year=2004|isbn=978-1-4020-2690-4}}.
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| [[Category:Module theory]]
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| [[Category:Multilinear algebra]]
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| [[Category:Homological algebra]]
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| [[Category:Binary operations]]
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