# Answer to Question #17650 in Abstract Algebra for Hym@n B@ss

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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