Question

Let R be a left artinian ring and C be a subring in the center Z(R) of R. Show that Nil C = C ∩ rad R.

Answer 1

Let a ∈Nil C. Then aR is a nil ideal of R, so aR ⊆rad R. This shows that Nil C ⊆ C ∩ rad R. For the converse, notethat rad R is nil (in fact nilpotent, since R is left artinian).Therefore, C ∩ rad R ⊆Nil C.