Let $M$ be a finitely presented $A$-module and $S \subset A$ be a multiplicative set. Hence have a surjective map $f:A^{\oplus n} \rightarrow M$
Let $S^{-1}M$ be a free $S^{-1}A$-module. Then we have an isomorphism $S^{-1}f:(S^{-1}A)^{\oplus n} \rightarrow S^{-1}M$.
Is it true that $S^{-1}$ker($f$)=Ker($S^{-1}f$)?
I think $"\subseteq"$ holds true but I am not sure about the other inclusion.