A finitely generated $S^{-1}R$-module isomorphic to the localization of a finitely generated $R$-module

Question

I am trying to prove the following statement:

Every finitely generated $S^{-1}R$-module, say $M$, is isomorphic to $S^{-1}N$, where $N$ is a finitely generated $R$-module ($S$ is an arbitrary multiplicative set).

My thoughts: the localization $S^{-1}N$ is a finitely generated $S^{-1}R$ module since $N$ is finitely generated.$\,$ Now I am looking for an isomorphism, say $\alpha$, between $S^{-1}N$ and $M$ such that for each generator of $S^{-1}N$,$\,$$\,$say $n_i$,$\,$ $\alpha(n_i)$ is a generator of $M$. But how can we tell that the cardinality of the generators set of $S^{-1}N$ is equal to the cardinality of the generators set of $M$. Any ideas?

Answer 1

Let $M$ be an $S^{-1}R$-module, generated by, say, $x_1,...,x_n$. Let $N$ be the sub-$R$-module of $M$ generated by $x_1,...,x_n$.

Clearly $N$ is finitely generated as an $R$-module, and there is a natural map of $S^{-1}R$-modules $S^{-1}N\to M$.

It will be surjective because its image contains $x_1,...,x_n$.

It will be injective because it factors as $S^{-1}N\to S^{-1}M\to M$, the first one is injective as $S^{-1}$ preserves injections, the second one is an isomorphism.

Therefore it is an isomorphism.