# Answer to Question #17653 in Abstract Algebra for Tsit Lam

Question #17653

For any left artinian ring R with Jacobson radical J, show that

soc(R_R) = {r ∈ R : Jr = 0} and soc(RR) = {r ∈ R : rJ = 0}.

soc(R_R) = {r ∈ R : Jr = 0} and soc(RR) = {r ∈ R : rJ = 0}.

Expert's answer

We use fact: soc(

Since

right analogue) to the modules

*M*)*⊆**{m**∈**M*: (rad*R*)*· m*= 0*},*withequality if*R/*rad*R*is an artinian ring.Since

*R/*rad*R*isartinian, the two desired equations follow by applying mentioned fact (and itsright analogue) to the modules

*RR*and*RR*.
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