Answer to Question #17653 in Abstract Algebra for Tsit Lam

Question

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}.

Answer 1

We use fact: soc(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 its
right analogue) to the modules RR and RR.