Question #17650

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

Expert's answer

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.

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