For me, the socle of an $R$-module $M$ is the unique maximal semisimple submodule of $M$. If $(R,\mathfrak m)$ is a local ring, this is equivalent to the maximal submodule annihilated by $\mathfrak m$. In the case of a local ring, I came across the statement
$$\operatorname{soc} M\cong\hom(R/\mathfrak m, M)_\mathfrak m,$$
understood as localisation of $R$-modules.
- I have difficulties to think of an example where the localisation of the hom-space along $\mathfrak m$ is necessary. Hence, I struggle to show this isomorphism.
- What would be the appropriate generalisation of this statement for non-local rings $R$?
