# Answer to Question #23564 in Algebra for Mohammad

Question #23564

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.

Expert's answer

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*.Need a fast expert's response?

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