# On the submodule $(0:_Ma)$ of a non Cohen-Macaulay module $M$

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.