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Answer to Question #24889 in Abstract Algebra for Melvin Henriksen

Question #24889
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.
Expert's answer
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.

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