Let $R$ be a ring, and $S$ a subring of $R$. Let $M$ be a right $R$-module, and $N$ a right $S$-submodule of $M$. If $N$ is flat (or faithfully flat) as a right $S$-module, does it then follow that the multiplication map$$N \otimes_S R \to NR \subseteq M$$is an isomorphism? Is flatness necessary for this to happen?
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