tensor product

Summary. The tensor productPlanetmathPlanetmathPlanetmath is a formal bilinearPlanetmathPlanetmath multiplication of two modules or vector spacesMathworldPlanetmath. In essence, it permits us to replace bilinear maps from two such objects by an equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath linear map from the tensor product of the two objects. The origin of this operationMathworldPlanetmath lies in classic differential geometry and physics, which had need of multiply indexed geometric objects such as the first and second fundamental formsMathworldPlanetmath, 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 AB, called the tensor product of A and B over R, together with a canonical bilinear homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath


distinguished, up to isomorphismMathworldPlanetmathPlanetmath, by the following universal propertyMathworldPlanetmath. Every bilinear R-module homomorphismMathworldPlanetmath


lifts to a unique R-module homomorphism


such that


for all aA,bB. Diagramatically:


The tensor product AB can be constructed by taking the free R-module generated by all formal symbols


and quotienting by the obvious bilinear relationsMathworldPlanetmathPlanetmath:

(a1+a2)b =a1b+a2b, a1,a2A,bB
a(b1+b2) =ab1+ab2, aA,b1,b2B
r(ab) =(ra)b=a(rb) aA,bB,rR


Basic . Let R be a commutative ring and L,M,N be R-modules, then, as modules, we have the following isomorphisms:

  1. 1.


  2. 2.


  3. 3.


  4. 4.


Definition (Categorical). Using the languagePlanetmathPlanetmath of categoriesMathworldPlanetmath, all of the above can be expressed quite simply by stating that for all R-modules M, the functorMathworldPlanetmath (-)M is left-adjoint to the functor Hom(M,-).

Title tensor product
Canonical name TensorProduct
Date of creation 2013-03-22 12:21:26
Last modified on 2013-03-22 12:21:26
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 12
Author rmilson (146)
Entry type Definition
Classification msc 13-00
Classification msc 18-00
Related topic Module
Related topic OuterMultiplication