Surjection $M\to R/P$

Question

Let $(R,\mathfrak{m})$ be a local Noetherian ring, and $M$ a finitely generated $R$-module. I am trying to show that there is a surjection $M\to R/P$ for any $P\in\operatorname{Supp} M$.

I know there is a surjection $M\to R/\mathfrak{m}$. But, it seems that this result doesn't help.

Need some help. Hope this question is not too obvious, but I am really running out of my tricks. Thanks.

Answer 1

Let $R=k[X,Y]_{(X,Y)}$ for a field $k$, $P=0$, $M=(X,Y)$ the maximal ideal of $R$. This admits no $R$-module epimorphism $M \to R$.