Answer to Question #24883 in Abstract Algebra for john.george.milnor

Question

Let N1(R) be the sum of all nilpotent ideals in a ring R. Show that N1(R) is a nil subideal of Nil*R which contains all nilpotent one-sided ideals of R.

Answer 1

Let N = N1(R).Note that any nilpotent ideal is in Nil*R, so N ⊆Nil*R, and N is nil. If A is a nilpotent (say) left ideal,then AR is a nilpotent ideal, so A ⊆AR ⊆ N.