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.
Given any r ∈ R, let s = ran−1∈ I. Then sa = 0, and a quick induction on mshows that (s + a)m= sm+asm−1 + a2sm−2+ · · · + am. Taking m = n and using sn= an= (s + a)n=0, wehave 0 = asn−1 + · · · + an−2s2+ an−1s = (at + 1)an−1sfor some t ∈ R. (Notethat s2 = ran−1s, etc.)Since (at)n+1 = a(ta)nt= 0, we have at + 1 ∈U(R), so 0 = an−1s= an−1ran−1, asdesired.