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.
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Expert's answer
2013-02-22T06:46:12-0500
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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