Let be a commutative ring with 1 and let be a free left unital -module of finite rank. Prove that , where means “isomorphic as -modules”.
Suppose is free on the set ; then every element of can be written uniquely as for some . Now define by . We claim that is an -module isomorphism.
Note that if and , then . Thus is an -module homomorphism.
Now suppose . Then . Since is free on the , we have for all , and thus . So , and thus is injective.
Finally, let . Define by . Evidently, is an -module homomorphism. Moreover, , so that . Hence is surjective.
Thus is an -module isomorphism, so that we have .