ABSTRACT
Proof: Suppose that I <I·A and IE £e. Then in view of Example 3.6.1 (i), I is a semiprime ring and so is A as well. Now there is an essential ideal K E e of/. By Propositions 3.7.6 and 3.7.7 we get that K as well as 1?3 are essential ideals of A. Since 1?3 <1 K E e, and e is hereditary, it follows that 1?3 E e. Thus A E £ e and the Proposition has been proved. 0
Now we can easily construct the semisimple class of the largest supernilpotent radical -y which excludes a given class e of semiprime rings (cf. Sands' construction Theorem 3.1.11). We define the hereditary closure operator 'H. by
THEOREM 3.7.9. Let e be any class of semiprime rings, and let a be the class of all subdirect sums of rings in £'He. Then Ua = U£'H.e and this is a supernilpotent radical. Moreover, a is the smallest semisimple class containing e and defining a supernilpotent upper radical.
