Suppose , , and are rings (with unity), is an -bimodule, and is an -bimodule. If we trace through our construction of the tensor product in the special case of extending scalars, we see that we can create all of the various constructions in a more general setting, ultimately forming a new module. In this new general setting, that module will be denoted and called the tensor product of and over . It will be an -bimodule and will satisfy a universal property similar to that of our previous construction.[1]. We describe the general construction here, but the vast majority of examples we consider will be less general.
The construction of
As a set, we first consider the Cartesian product of the elements of the underlying sets of and , i.e., we consider the set . We then form the free -module on this set, obtaining the abelian group whose elements consist of all formal finite sums of the form where , and . We then prepare to "recover" the lost additive structures of and by letting denote the subgroup of this abelian group generated by all elements of the form
with and
with and
As well as all elements of the form
for , , and
This last set of elements is to recover the right -action on and left -action on .
The resulting quotient group is then denoted . If we write for the coset represented by the pair in this quotient group (and extend that notation linearly), then every element in can be represented (non-uniquely!) by a finite sum of the form for some and .[2]
How about the -bimodule structure? You could probably guess it, in that for simple tensors we define and for and , and then extend to all tensors linearly.[3]
Basic properties of tensors
It's worth listing for easy reference the basic properties of simple tensors, as the construction was tailored especially for to have these properties:
for every and
for every and
for every , , and
for every , , and
for every , , and
Special cases
Tensor products of abelian groups
Suppose and are abelian groups. We have seen that and then also have unique -bimodule structures, and so we can form their tensor product . This is another -bimodule, i.e., abelian group.
Extension of scalars
Suppose is an -module and is a subring of . We have seen that we have a natural -bimodule structure on and -bimodule structure on , so we can form the tensor product . The result is an -bimodule, i.e., a left -module. Comparing the construction above with our previous construction, we can see that this the same -module we constructed (and also denoted ) previously.
Tensor product of left -modules over a commutative ring
Suppose is a commutative ring and and are left -modules. By taking the standard -module structure (i.e., -bimodule structure) on and the canonical -bimodule structure on , we can form the tensor product . The result is an -bimodule, i.e., a left -module.
We can also consider the standard -bimodule structures on both and and form the tensor product , which is now an -bimodule. It is a bit unfortunate that this notation is identical to the one above, and is the first situation in which one should be careful to distinguish the type of modules being considered for the tensor product construction.
Tensor products of rings
Suppose and are rings (with unity). We can then also view and as -bimodules, by forgetting their internal multiplicative operations and remembering only their additive structures.[4] We can then form the -bimodule . We can give this abelian group the structure of a ring by defining the product on simple tensors by
and then extending linearly to all of . One can verify that this makes into a ring (with unity given by ).
This tensor product construction on rings is usually simply denoted .