Let $i:M\to$ $N$ be an injection and let $f:M \to M/N$ is the canonical epimorphism, why $i_S$, $f_S$ are injective, surjective respectively?

Question

Let $N$ be a submodule of the $R$-module $M$. Let $i:M\to$ $N$ be an injection. Let $N_S$, $M_S$ be the localizations of $N$, $M$ at the submonoid $S$ of the multiplicative monoid of $R$ respectively.

I am not understanding why $i_S:M_S\to$$N_S$ is injective. I think that since $i$ is injective, then $ker(i)=0$. I cannot show why $ker(i_S)=0$.

Moreover, if $f:M\to$M/N is the canonical homomorphism, why $f_S:M_S\to$ $M_S/N_S$ is surjective?

Would you help me please? Thank you in advance.

Answer 1

Let $ a/s \in M_S$ such that $ i_S(a/s) = 0 \implies i(a)/s = 0 \implies \exists t \in S $ such that $ i(a).t = 0 \implies i(at) = 0 \implies at = 0 \implies a/s = 0 $. The surjectivity also has a similar argument.