Question

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.

Answer 1

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