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Let $(R,\mathfrak m)$ be a Noetherian local ring such that $(0:_R \mathfrak m)\neq 0$. Then, is it true that for every non-zero ideal $I$ of $R$, one also has $I\cap (0:_R \mathfrak m)\neq 0$ ?

I know this is true when $R$ is Artinian, but not sure otherwise.

Please help.

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    $\begingroup$ $R = k[[x,y]]/(x^2,xy)$, $I = (y)$ $\endgroup$
    – math54321
    Mar 18, 2023 at 23:55

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