Does the torsion submodule and the $0$-th local Cohomology module coincide over local Cohen-Macaulay ring?

Question

Let $M$ be a finitely generated module over a local Cohen-Macaulay ring $(R,\mathfrak m)$.

If $x\in M$ is annihilated by a non-zero-divisor $r\in \mathfrak m$ , then is it true that $\mathfrak m^n x=0$ for some $n\ge 1$?

This is definitely true if $\dim R=1$. Indeed, then the ideal $(r)$ has height $1$, so is $\mathfrak m$-primary, hence $\mathfrak m^n \subseteq (r)$ for some $n\ge 1$.

I am pretty sure this need not be true in higher dimension , but I cannot come up with counterexample.

Please help.

Answer 1

It is not true is higher dimensions. Consider the following counterexample.

Take $R=k[x, y]_{(x, y)}$ the local ring of affine plane at the origin.

Take $M=R/(x^2)$.

Now $x\in M$ is killed by $x\in \mathfrak{m}=(x, y)R$, but $\mathfrak{m}^nx\neq 0$ for any $n\geq 1$, since $y^nx\neq 0$.