Modules isomorphic over subring

Question

Let $R$ be a commutative ring with identity. Let $M,N$ be two isomorphic $R$-modules, and let $S \subset R$ be a subring of $R$, sharing a common identity element. Is it then true that $M$ and $N$ are isomorphic as $S$-modules? My intuition says yes, and I think it's proven by simply stating that an $R$-module homomorphism $\varphi : M \to N$ is also an $S$-module homomorphism, however, for some reason it feels like I'm missing something.

Answer 1

You are not missing anything. If $\phi:M\rightarrow N$ is an $R$-module isomorphism, then $\phi$ preserves addition and for every $s\in S$ we have

$$\phi(s\cdot m) = s\cdot \phi(m)$$

because $S\subseteq R$. Thus $\phi$ is a morphism of $S$-modules. Moreover, $\phi$ is bijective. Hence it is an isomorphism of $S$-modules. This holds even for noncommutative rings and left modules.