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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.
Expert's answer
The case of the Jacobson radical isclear. The case of the lower nilradical follows easily from the interpretation
of Nil*R as the smallest semiprime ideal of R. Now consider theupper nilradical Nil*R. If N ⊇Nil*R isan ideal with N2 ⊆Nil*R, then N isclearly nil, and so N = Nil*R. This checks that Nil*R issemiprime

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