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 contains a nonzero nilpotent left ideal, and R has a nonzero nilpotent ideal.

Answer 1

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.