Let be a commutative ring with ideals and . Let and be -modules (in fact -bimodules) in the usual way.
- Prove that every element of can be written as a simple tensor of the form .
- Prove that .
We will prove the first result first for simple tensors; the extension to arbitrary sums of tensors follows by tensor distributivity. Let be an arbitrary simple tensor in . Now , as desired.
Now define by . First, suppose and . Then and , so that . Thus is well-defined. It is clear that is -balanced, and so induces an -module homomorphism . Since , is surjective. Now suppose is in the kernel of ; then . Say where and . Then , and hence is an isomorphism.