Let be a commutative ring and let be a free -module of finite rank ; say is a basis for .
- Let be any nonzero -module. (Since is commutative, is naturally an -bimodule.) Show that . In particular, every element of can be written uniquely in the form .
- Show that if , where the are merely assumed to be -linearly independent in , it need not be the case that the are all zero. (That is, it is crucial that the generate as an -module.)
Define a mapping by . (This is well defined since is a basis for .) Moreover, , , and , so that is bilinear. By the universal property of tensor products, induces a unique group homomorphism (indeed, -module homomorphism) such that .
Now define by . Clearly is an -module homomorphism.
Moreover, note that . Similarly, . Thus and are mutual inverses, and we have .
In particular, if and only if for all .
Now for the counterexample, let , let be the free -module of rank 1, and let be a -module in the usual way. Consider the simple tensor in ; note that , but that .