Addendum A to 262

[!Proposition]
Let $R$ be a Noetherian ring and $M$ be a module over $R$. Let $\mathfrak{p}\subseteq R$ be a prime ideal. Then, $\mathfrak{p}=\text{Ann}(m)$ for some $m\in M$ $⟺$ there is an injection $R/\mathfrak{p}\hookrightarrow M$.

[!Proof]-

$(⟹)$ Consider the $R$-module homomorphism $\phi :R\to M$ defined by $r\mapsto rm$. Clearly, $\ker \phi =\mathfrak{p}$. By