Tensor product of modules

is an abelian group together with a balanced product (as defined above)

The universal property of a tensor product has the following important consequence:

(This section need to be updated. For now, see § Properties for the more general discussion.)

It is possible to extend the definition to a tensor product of any number of modules over the same commutative ring. For example, the universal property of

The tensor product, in general, does not commute with inverse limit: on the one hand,

The structure of a tensor product of quite ordinary modules may be unpredictable.

Tensor products can be applied to control the order of elements of groups. Let G be an abelian group. Then the multiples of 2 in

Another useful family of examples comes from changing the scalars. Notice that

is balanced, and the subgroup has been chosen minimally so that this map is balanced. The universal property of ⊗ follows from the universal properties of a free abelian group and a quotient.