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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 ?

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

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  • $\begingroup$ how do we show that $k[x,y]$ and $ (x,y)$ are not isomorphic as $k[x,y]$ modules ? $\endgroup$ – user228168 Oct 31 '16 at 15:23
  • $\begingroup$ Show that one of them cannot be generated by one element. $\endgroup$ – Mariano Suárez-Álvarez Oct 31 '16 at 16:52

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