About a short proof of Krull principal ideal theorem

Question

How from this theorem I can get a proof of Krull principal ideal theorem?

I understand that w.l.g. we can prove it for a Noetherian local ring. But why we can consider that $(x)$ is $M$-primary?

Answer 1

Let $P$ be a minimal prime over $(x)$. Then $PR_P$ is minimal over $xR_P$. The ring $R_P$ is local and $xR_P$ is $PR_P$-primary.