Let $R$ be a commutative ring and let $A$ and $B$ be $R$-algebras. If $f: A\to B$ is a homomorphism of $R$-algebras, such that one can show that $f$ is an isomorphism of the underlying $R$-modules, is $f$ an isomorphism of $R$-algebras?
I would say that if $f$ is a homomorphism of $R$-algebras and we know that $f$ is an isomorphism of the underlying abelian groups of $A$ and $B$, that should suffice to conclude that $f$ is an isomorphism of $R$-algebras because $f$ would be a bijective algebra homomorphism which is the definition of an isomorphism of algebras.