ABSTRACT
Radical Theory for Associative Rings 79
If 'Y is any supernilpotent radical, then its semisimple class S-y is hereditary (Corollary 3.1.4), has only semiprime rings (Proposition 3.6.3), is closed under essential extensions (Proposition 3.2.6), and therefore S-y is a weakly special class. 0
EXAMPLE 3.7.13. The class of all semiprime rings is a weakly special class. By definition the semisimple class S{3 of the Baer radical {3 consists of semiprime rings. Further, if A ¢ S{3, then {3(A) -:/: 0 and in view of Corollary 3.4.8 there exists an ideal I of [3(A} such that I -:/: 0 and I 2 = 0. Hence by the Andrunakievich Lemma the ideal 1 of A generated by I is nilpotent and 1-:/: 0. Thus A is not semiprime, and therefore S{3 coincides with the class of all semiprime rings, and the Baer radical {3 is the upper radical of that class. Since {3 is hereditary, S{3 is closed under essential extensions (Proposition 3.2.6). Moreover, S{3 is hereditary (Corollary 3.1.4), and so S {3 is a weakly special class, in fact the largest weakly special class.
