Let $(R,\mathfrak m)$ be a local Cohen-Macaulay ring of positive dimension. Let $M$ be a finitely generated maximal Cohen-Macaulay module i.e. $\operatorname{depth} M=\dim R$. If $\operatorname{Supp}(M)=\operatorname{Spec}(R)$, then is it true that $M$ is faithful?
My thoughts: Since $\operatorname{Supp}(M)=\operatorname{Spec}(R)$, so the radical of the annihilator ann$_R(M)$ is exactly the nilradical of $R$, so every element in the annihilator of $M$ is nilpotent. We would be done if $R$ were reduced, but that's not necessarily the case.
Please help.
Let $R=K[X,Y]/(X^2Y)$ and $M=R/(xy)$. Then $M$ is MCM, $\operatorname{Ann}_R(M)=(xy)$, and $\operatorname{Supp}(M)=\operatorname{Spec}(R)$ since every prime ideal of $R$ contains $x$ or $y$.