Answer to Question #24890 in Abstract Algebra for Melvin Henriksen

Question

Let I be a left ideal in a ring R such that, for some integer n, an = 0 for all a ∈ I. Show that I ⊆ Nil*R.

Answer 1

Let J = Nil*R. If theimage I of I in R/J is nonzero, then there is a nonzeronilpotent ideal in R/J, which is impossible. Therefore, we must have I= 0, that is, I ⊆Nil*R.