tensor product
Summary. The tensor product^{} is a formal bilinear^{} multiplication of two modules or vector spaces^{}. In essence, it permits us to replace bilinear maps from two such objects by an equivalent^{} linear map from the tensor product of the two objects. The origin of this operation^{} lies in classic differential geometry and physics, which had need of multiply indexed geometric objects such as the first and second fundamental forms^{}, and the stress tensor — see Tensor Product (Classical) (http://planetmath.org/TensorProductClassical).
Definition (Standard). Let $R$ be a commutative ring, and let $A,B$ be $R$modules. There exists an $R$module $A\otimes B$, called the tensor product of $A$ and $B$ over $R$, together with a canonical bilinear homomorphism^{}
$$\otimes :A\times B\to A\otimes B,$$ 
distinguished, up to isomorphism^{}, by the following universal property^{}. Every bilinear $R$module homomorphism^{}
$$\varphi :A\times B\to C,$$ 
lifts to a unique $R$module homomorphism
$$\stackrel{~}{\varphi}:A\otimes B\to C,$$ 
such that
$$\varphi (a,b)=\stackrel{~}{\varphi}(a\otimes b)$$ 
for all $a\in A,b\in B.$ Diagramatically:
$$\text{xymatrix}\text{ar}{[dr]}^{(}.55)\varphi \text{ar}{[r]}^{\otimes}A\times B\mathrm{\&}A\otimes B\text{ar}\mathrm{@}>{[d]}^{(}.4)\exists !\stackrel{~}{\varphi}\mathrm{\&}C$$ 
The tensor product $A\otimes B$ can be constructed by taking the free $R$module generated by all formal symbols
$$a\otimes b,a\in A,b\in B,$$ 
and quotienting by the obvious bilinear relations^{}:
$({a}_{1}+{a}_{2})\otimes b$  $={a}_{1}\otimes b+{a}_{2}\otimes b,$  ${a}_{1},{a}_{2}\in A,b\in B$  
$a\otimes ({b}_{1}+{b}_{2})$  $=a\otimes {b}_{1}+a\otimes {b}_{2},$  $a\in A,{b}_{1},{b}_{2}\in B$  
$r(a\otimes b)$  $=(ra)\otimes b=a\otimes (rb)$  $a\in A,b\in B,r\in R$ 
Note.
Basic . Let $R$ be a commutative ring and $L,M,N$ be $R$modules, then, as modules, we have the following isomorphisms:

1.
$R\otimes M\cong M$,

2.
$M\otimes N\cong N\otimes M$,

3.
$(L\otimes M)\otimes N\cong L\otimes (M\otimes N)$

4.
$(L\oplus M)\otimes N\cong (L\otimes N)\oplus (M\otimes N)$
Definition (Categorical). Using the language^{} of categories^{}, all of the above can be expressed quite simply by stating that for all $R$modules $M$, the functor^{} $()\otimes M$ is leftadjoint to the functor $\mathrm{Hom}(M,)$.
Title  tensor product 

Canonical name  TensorProduct 
Date of creation  20130322 12:21:26 
Last modified on  20130322 12:21:26 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  12 
Author  rmilson (146) 
Entry type  Definition 
Classification  msc 1300 
Classification  msc 1800 
Related topic  Module 
Related topic  OuterMultiplication 