Localization of a $\mathfrak p$-primary submodule

Question

Let $\mathfrak p\in\textrm{Spec}A.$ Let $N$ be a $\mathfrak p$-primary submodule of $A$-module $M$ and let $S\subset A$ be a multiplicative system. Show that:

1. If $S\cap\mathfrak p\ne\varnothing$ then $N_S=M_S.$

2. Is $S\cap\mathfrak p=\varnothing$ then $N_S$ is a $\mathfrak pA_S$-primary in $M$.

Here are my thoughts: $\sqrt{\textrm{Ann}(M/N)}=\mathfrak p.$ Let $x\in S\cap\mathfrak p.$ Then $x^k\in\textrm{Ann(M/N)}$, i.e. for all $m\in M$ $x^k(m+N)=0,$ hence, $x^k\in N$. Let $y\in M_S.$ Then $y=\frac{m}{s}$. I think we must represent it somehow using $x^k$ but have got no idea how.

In order to show $N_S$ is a $\mathfrak pA_S$-primary in $M$ we must prove that $\mathfrak pA_S=\sqrt{\textrm{Ann}(M/N)}$. I do not how.

Can you please help me?

Answer 1

For the 1st part: $y=m/s=x^km/x^ks$ Now $x^km\in N$ (since $x\in p=\sqrt{Ann(M/N)}$ ) and $x^k s\in S$.

For the 2nd part: to prove $pA_S=\sqrt{Ann(M_S/N_S)}$.

Let $x/s\in pA_S$. Now let $m/t\in M_S$. $x^nm\in N$ for some $n$. So $(x/s^n)m/t=x^nm/s^nt\in N_S$. So $x/s\in pA_S$.

Conversely, I will further assume that $M$ is a finitely generated $ A-$module. If $x/s\in \sqrt{Ann(M_S/N_S)}$ then $x^n/s^nM_S\subset N_S$. So $x^n/s^nm_i \subset N_S$ where $m_i$’s are generators for $M$. So for some $t\in S$, $ x^ntm_i \in N$. Thus $x^ntM\subset N$. So $(x^nt)^r\in p$. Hence $x\in p$.