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The more general form of Krull intersection theorem says:

Let $R$ be local and Noetherian and $I \subset R$ a proper ideal. If $M$ is finitely generated over $R$, and $N=\cap_1^{\infty} I^iM$, then $IN=N$.

What is the simplest counter-examples when one (and only one) condition among: $R$ local, $R$ Noetherian or $M$ finitely generated is dropped? So this is three questions I guess.

Sorry if this is too easy for this site. It has been a while, you know!

LATER: Andrea's answer gave a counter-example to the stronger statement: there is an element $r \in I$ such that $N(1-r)=0$. I believe it is not a counter-example to the form stated above, see David Eisenbud's book on commutative algebra, the Example after Corollary 5.5 and Exercise 5.6.

However, Dustin Cartwright pointed out that one can safely drop the "local" hypothesis. So there are only two questions left.

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Counter-examples to Krull's intersection theorem

The more general form of Krull intersection theorem says:

Let $R$ be local and Noetherian and $I \subset R$ a proper ideal. If $M$ is finitely generated over $R$, and $N=\cap_1^{\infty} I^iM$, then $IN=N$.

What is the simplest counter-examples when one (and only one) condition among: $R$ local, $R$ Noetherian or $M$ finitely generated is dropped? So this is three questions I guess.

Sorry if this is too easy for this site. It has been a while, you know!