Tensor of Abelian Groups

Tensor Product of Abelian Groups

Let . Then the tensor product is defined as the quotient of the free abelian group on with the following relations

Identity in the monoid

Let and define a map with

Let be any representative of in . If , then , so WMA . For all , we have

To show this a homomorphism, let . We have . A moment’s thought shows the inverse is a homomorphism, and both and are identities. Thus is an isomorphism and

Tensor product of abelian group and

Let be an abelian group. For , we can write

Using bilinearity in the second component, any element in the tensor product can be written in the form . Mapping this to , we see that