Let be a commutative ring with 1 and let be a unital subring. Prove that as -algebras, and are isomorphic.
Define by . This mapping is certainly -bilinear, and so (since is commutative) induces an additive group homomorphism such that . Note that , so that is a ring homomorphism. Moreover, we have and , so that is an -algebra homomorphism.
Since , is surjective. Note that every simple tensor (hence every element) of can be written in the form . If , then we have for all , and thus . In particular, , so that is injective.
Thus and are isomorphic as -algebras.