# Answer to Question #24888 in Abstract Algebra for Melvin Henriksen

Question #24888

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.

Expert's answer

Given any

*r**∈**R*, let*s*=*ra*^{n−}^{1}*∈**I*. Then*sa*= 0, and a quick induction on*m*shows that (*s*+*a*)*=*^{m}*s*+^{m}*as*^{m−}^{1}+*a*^{2}*s*^{m−}^{2}+*· · ·*+*a*Taking^{m}.*m*=*n*and using*s*=^{n}*a*= (^{n}*s*+*a*)*=0, wehave 0 =*^{n}*as*^{n−}^{1}+*· · ·*+*a*^{n−}^{2}*s*^{2}+*a*^{n−}^{1}*s*= (*at*+ 1)*a*^{n−}^{1}*s*for some*t**∈**R*. (Notethat*s*^{2}=*ra*^{n−}^{1}*s*, etc.)Since (*at*)^{n}^{+1}=*a*(*ta*)*= 0, we have*^{n}t*at*+ 1*∈*U(*R*), so 0 =*a*^{n−}^{1}*s*=*a*^{n−}^{1}*ra*^{n−}^{1}*,*asdesired.
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