Let $(R,\mathfrak m)$ be a local ring. Then $\mathrm{soc}(M) = \{ r \in M \mid \mathfrak mr = 0 \}$.

Question

Given that $(R,\mathfrak m)$ is a local ring we want to show that $\mathrm{soc}(M) = \{ r \in M \mid \mathfrak mr = 0 \}$, where $M$ is an $R$-module.

I tried the following. Let N be an essential module of R. then I tried showing that $ soc(M) \subset N$ But I was unable to do this. any ideas? thanks in advance