Let be a ring with 1 and let be a left unital -module. Prove that . (We use to say “isomorphic as -modules”.)
Define by . We claim that is an -module isomorphism.
Let and . Then . Thus is an -module homomorphism.
Suppose now that . Then for all , . Hence , and so is injective.
Now let . Define by . Since for all and we have , is an -module homomorphism, and so . Moreover, . Thus is surjective.
Thus we have .