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*^{m}. Taking *m *= *n *and using *s*^{n}= *a*^{n} = (*s *+ *a*)^{n} =0, wehave 0 = *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*)^{n}t= 0, we have *at *+ 1 *∈*U(*R*), so 0 = *a*^{n−}^{1}*s*= *a*^{n−}^{1}*ra*^{n−}^{1}*, *asdesired.

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