Answer to Question #17651 in Abstract Algebra for Hym@n B@ss

Question

Show that for any ring R, soc(R) is an ideal of R.

Answer 1

For any minimal left ideal I ⊆ R and any r ∈ R, Ir is a homomorphic image of RI,so Ir is either 0 or another minimal left ideal. Since soc(RR)=(sum) I, we see that soc(RR) is an ideal.