Let be any (left) -module. Fix and let be a positive integer. Prove that given by is a well-defined -module homomorphism if and only if . Prove that . (See this previous exercise for a definition of .)

Suppose that is a well-defined -module homomorphism. In particular, we have . So . Now suppose . If mod , we have . Say . Now , so that . Thus , and so . So is well-defined. Now suppose and . Then . So is a -module homomorphism.

Now define by . This mapping is properly defined by the above argument. We claim that is a -module isomorphism.

First we show that is a -module homomorphism. To that end, let and . Now . Thus we have , and so is a -module homomorphism.

Now suppose . Then for all ; in particular, . Thus , and so is injective.

Finally, let , and let . Note that , so that . We claim that . To see this, note that for all we have . Thus , and so is surjective.

So we have .