Answer to Question #24888 in Abstract Algebra for Melvin Henriksen

Question

Let I be a left ideal in a ring R such that, for some integer n ≥ 2, an = 0 for all a ∈ I. Show that an−1Ran−1 = 0 for all a ∈ I.

Answer 1

Given any r ∈ R, let s = ra^n−1∈ I. Then sa = 0, and a quick induction on mshows that (s + a)^m = s^m +as^m−1 + a²s^m−2+ · · · + a^m. Taking m = n and using sⁿ= aⁿ = (s + a)ⁿ =0, wehave 0 = as^n−1 + · · · + a^n−2s²+ a^n−1s = (at + 1)a^n−1sfor some t ∈ R. (Notethat s² = ra^n−1s, etc.)Since (at)ⁿ⁺¹ = a(ta)ⁿt= 0, we have at + 1 ∈U(R), so 0 = a^n−1s= a^n−1ra^n−1, asdesired.