Possible way to base change to a module?

Question

Let $M$ be an $R$-module (all rings are commutative). Let $S$ be a subring of $R$. Is there anyway to figure out what are all the possible $S$-module $N$ such that $N\otimes_{S}R\cong M$? In particular, what happened if $R^{\times}\bigcap S=S^{\times}$?