# Answer to Question #24890 in Abstract Algebra for Melvin Henriksen

Question #24890

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 ⊆ Nil*R.

Expert's answer

Let

*J*= Nil**R*. If theimage*I*of*I*in*R/J*is nonzero, then there is a nonzeronilpotent ideal in*R/J*, which is impossible. Therefore, we must have*I*= 0, that is,*I**⊆*Nil**R*.
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