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 
,
 |
(1)
|
 |
(2)
|
 |
(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 
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
 |
(4)
|
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 
.
