Tensor Product of Modules - Definition

Definition

Let M,N and R be as in the previous section. The tensor product over R

is an abelian group together with a bilinear map (in the sense defined above)

which is universal in the following sense:

For every abelian group Z and every bilinear map

there is a unique group homomorphism

such that

As with all universal properties, the above property defines the tensor product uniquely up to a unique isomorphism: any other object and bilinear map with the same properties will be isomorphic to M ⊗_R N and ⊗. The definition does not prove the existence of M ⊗_R N; see below for a construction.

The tensor product can also be defined as a representing object for the functor Z → Bilin_R(M,N;Z). This is equivalent to the universal mapping property given above.

Strictly speaking, the ring used to form the tensor should be indicated: most modules can be considered as modules over several different rings or over the same ring with a different actions of the ring on the module elements. For example, it can be shown that R ⊗_R R and R ⊗_Z R are completely different from each other. However in practice, whenever the ring is clear from context, the subscript denoting the ring may be dropped.