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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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