Question

Show that soc(M) ⊆ {m ∈ M : (rad R) • m = 0}, with equality if R/rad R is an artinian ring.

Answer 1

The first conclusion follows fromthe fact that (rad R)V = 0 for any simple left R-module V. Now assume R/rad R is artinian. Let N = {m ∈ M : (rad R) · m = 0}, whichis an R-submodule of M. Viewing N as a module over thesemisimple ring R/rad R, we see that RN is semisimple.Therefore, N ⊆ soc(M), as desired.