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Let $(R,m)$ denote a commutative Noetherian local ring and M a finitely generated R-module of dimension $d$. We say M is a Cohen-Macaulay R-module, if $dim_RM=depth_RM=d$. For every $r\in R$ let $(0:_Mr):= \{x\in M ;rx=0\}$.

Let $a\in \frak m$ such that $(0:_Ma)=(0:_Ma^2)$ and $M/a^2M$ is a Cohen-Macaulay $R$-module of dimension $d-1$. If $M$ is not Cohen-Macaulay can we conclude that $(0:_Ma)\neq0$?

It should be mentioned that $\phi:M/(0:_Ma^2)\longrightarrow a^2M$ via $x+(0:_Ma^2)\rightarrow a^2x$ is an $R$-module isomorphism.

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