$M$ be a finitely generated module over commutative unital ring $R$ , $N,P$ submodules , $P\subseteq N \subseteq M$ and $M\cong P$ , is $M\cong N$?

Question

Let $R$ be a commutative ring with unity , $M$ be a finitely generated module over $R$ , let $N,P$ be submodules of $M$ such that $P\subseteq N \subseteq M$ and $M\cong P$ , then is it true that $M\cong N$ ? If not true , then what happens if we also assume that $M$ is Noetherian ?

Answer 1

Take $R=k[x,y]$ with $k$ a field, $M=R$, $P=(xy)$ and $N=(x,y)$