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Module Tensor Product


The tensor product between modules A and B is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring R. The familiar formulas hold, but now alpha is any element of R,

 (a_1+a_2) tensor b=a_1 tensor b+a_2 tensor b
(1)
 a tensor (b_1+b_2)=a tensor b_1+a tensor b_2
(2)
 alpha(a tensor b)=(alphaa) tensor b=a tensor (alphab).
(3)

This generalizes the definition of a tensor product for vector spaces since a vector space is a module over the scalar field. Also, vector bundles can be considered as projective modules over the ring of functions, and group representations of a group G can be thought of as modules over CG. The generalization covers those kinds of tensor products as well.

There are some interesting possibilities for the tensor product of modules that don't occur in the case of vector spaces. It is possible for A tensor _RB to be identically zero. For example, the tensor product of C_2 and C_3 as modules over the integers, C_2 tensor _ZC_3, has no nonzero elements. It is enough to see that a tensor b=0. Notice that 1=3-2. Then

 (1)a tensor b=(3-2)a tensor b=(-2a) tensor b+a tensor (3b)=0+0=0,
(4)

since -2a=-a-a=0 in C_2 and 3b=b+b+b=0 in C_3. In general, it is easier to show that elements are zero than to show they are not zero.

Another interesting property of tensor products is that if f:A->B is a surjection, then so is the induced map g:A tensor C->B tensor C for any other module C. But if f:A->B is injective, then g:A tensor C->B tensor C may not be injective.

For example, f:C_2->C_4, with f(1)=2 is injective, but g:C_2 tensor _ZC_2->C_4 tensor _ZC_2, with g(1 tensor 1)=2 tensor 1, is not injective. In C_4 tensor _ZC_2, we have 2 tensor 1=1 tensor 2=1 tensor 0=0.

There is an algebraic description of this failure of injectivity, called the tor module.

Another way to think of the tensor product is in terms of its universal property: Any bilinear map from A×B:->C factors through the natural bilinear map A×B->A tensor B.


See also

Group Representation, Module, Module Direct Sum, Projective Module, Representation Tensor Product, Tor, Universal Property, Vector Bundle, Vector Space, Vector Space Tensor Product

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Module Tensor Product." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ModuleTensorProduct.html

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