# Answer to Question #24884 in Abstract Algebra for john.george.milnor

Question #24884

Let N1(R) be the sum of all nilpotent ideals in a ring R. If N1(R) is nilpotent, show that N1(R) = Nil*R.

Expert's answer

Let

*N*=*N*1(*R*).Say*Nn*= 0. Then*R/N*has no nonzero nilpotent ideals. (For, if*I/N*is a nilpotent ideal of*R/N*, then*Im**⊆**N*for some*m*, and hence*Imn**⊆**Nn*= 0. This implies that*I**⊆**N*.) Therefore,*N*is semiprime, so*N*=Nil**R*.
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