# Answer to Question #24904 in Abstract Algebra for Mohammad

Question #24904

Let Rad R denote one of the two nilradicals, or the Jacobson radical, or the Levitzki radical of R.

Show that Rad R is a semiprime ideal.

Show that Rad R is a semiprime ideal.

Expert's answer

The case of the Jacobson radical isclear. The case of the lower nilradical follows easily from the interpretation

of Nil*

of Nil*

*R*as the smallest semiprime ideal of*R*. Now consider theupper nilradical Nil**R*. If*N**⊇*Nil**R*isan ideal with*N*2*⊆*Nil**R*, then*N*isclearly nil, and so*N*= Nil**R*. This checks that Nil**R*issemiprime
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