# 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.

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*a*^{n−}^{1}*<>*0. Then*a*^{n−}^{1}*Ra*^{n−}^{1}=0, so (*Ra*^{n−}^{1}*R*)^{2}= 0. Therefore*Ra*^{n−}^{1}*R*is a nonzero nilpotent ideal, and*I*contains the nonzero nilpotentleft ideal*Ra*^{n−}^{1}.
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