Radical of a ring: Difference between revisions

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In [[mathematics]], the [[tensor product]] of two [[algebra (ring theory)|''R''-algebras]] is also an ''R''-algebra. This gives us a '''tensor product of algebras'''. The special case ''R'' = '''Z''' gives us a '''tensor product of rings''', since rings may be regarded as '''Z'''-algebras.
 
Let ''R'' be a [[commutative ring]] and let ''A'' and ''B'' be [[algebra (ring theory)|''R''-algebras]]. Since ''A'' and ''B'' may both be regarded as [[module (mathematics)|''R''-modules]], we may form their [[tensor product of modules|tensor product]]
:<math>A \otimes_R B</math>
which is also an ''R''-module. We can give the tensor product the structure of an algebra by defining<ref>Kassel (1995), {{Google books quote|id=S1KE_pToY98C|page=32|text=we put an algebra structure on the tensor product|p. 32}}.</ref>
:<math>(a_1\otimes b_1)(a_2\otimes b_2) = a_1a_2\otimes b_1b_2</math>
and then extending by linearity to all of {{nowrap|''A''&thinsp;⊗<sub>''R''</sub>&thinsp;''B''}}. This product is easily seen to be ''R''-bilinear, associative, and unital with an identity element given by {{nowrap|1<sub>''A''</sub>&thinsp;⊗&thinsp;1<sub>''B''</sub>}},<ref>Kassel (1995), {{Google books quote|id=S1KE_pToY98C|page=32|text=Its unit is|p. 32}}.</ref> where 1<sub>''A''</sub> and 1<sub>''B''</sub> are the identities of ''A'' and ''B''. If ''A'' and ''B'' are both commutative then the tensor product is as well.
 
The tensor product turns the [[category (mathematics)|category]] of all ''R''-algebras into a [[symmetric monoidal category]].  
 
There are natural homomorphisms of ''A'' and ''B'' to {{nowrap|''A''&thinsp;⊗<sub>''R''</sub>&thinsp;''B''}} given by<ref>Kassel (1995), {{Google books quote|id=S1KE_pToY98C|page=32|text=get algebra morphisms|p. 32}}.</ref>
:<math>a\mapsto a\otimes 1_B</math>
:<math>b\mapsto 1_A\otimes b</math>
These maps make the tensor product a [[coproduct]] in the category of commutative ''R''-algebras. (The tensor product is ''not'' the coproduct in the category of all ''R''-algebras. There the coproduct is given by a more general [[free product]] of algebras).
 
The tensor product of algebras is of constant use in [[algebraic geometry]]: working in the [[opposite category]] to that of commutative ''R''-algebras, it provides [[pullback (category theory)|pullback]]s of [[affine scheme]]s, otherwise known as [[fiber product]]s.
 
<!--Stuf removed from [[ring (mathematics)]]:
Since any ring is both a left and right [[module (mathematics)|module]] over itself, it is possible to construct the [[tensor product of rings|tensor product]] of ''R'' over a ring ''S'' with another ring ''T'' to get another ring provided ''S'' is a central subring of ''R'' and ''T''.
-->
 
==See also==
*[[Extension of scalars]]
*[[Tensor product of modules]]
*[[Tensor product of fields]]
*[[Linearly disjoint]]
*[[Multilinear subspace learning]]
 
== Notes ==
<references/>
 
== References ==
* {{Citation| last1=Kassel| first1=Christian| date=1995| title=Quantum groups| volume = 155| series=Graduate texts in mathematics | publisher=Springer| isbn = 978-0-387-94370-1}}.
 
{{DEFAULTSORT:Tensor Product Of Algebras}}
[[Category:Algebras]]
[[Category:Ring theory]]
[[Category:Commutative algebra]]
[[Category:Multilinear algebra]]

Latest revision as of 23:55, 28 November 2014

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