The tensor product between modules and is a more general notion than the vector space tensor product. In this case, we replace "scalars" by a ring . The familiar formulas hold, but now is any element of ,
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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 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 to be identically zero. For example, the tensor product of and as modules over the integers, , has no nonzero elements. It is enough to see that . Notice that . Then
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since in and in . 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 is a surjection, then so is the induced map for any other module . But if is injective, then may not be injective.
For example, , with is injective, but , with , is not injective. In , we have .
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 factors through the natural bilinear map .