Answer to Question #24889 in Abstract Algebra for Melvin Henriksen
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 contains a nonzero nilpotent left ideal, and R has a nonzero nilpotent ideal.
We may assume n is chosenminimal. Since I <> 0, n ≥ 2. Fix an element a ∈ I with an−1<>0. Then an−1Ran−1 =0, so (Ran−1R)2 = 0. Therefore Ran−1Ris a nonzero nilpotent ideal, and I contains the nonzero nilpotentleft ideal Ran−1.