Left hereditary supernilpotent radical classes

Abstract

In [1] the concept of hereditary left weakly special class of rings was introduced and, with the aid of this class, a necessary and sufficient condition for a supernilpotent radical class to be left hereditary was given. In this note we begin with a construction of the smallest hereditary left special (weakly special) class of rings which contains a given hereditary class M of prime (semi-prime) rings. We shall call this class, which is described in Theorem 2, the hereditary left special (weakly special) closure of the given class. This class is then used to derive criteria equivalent to UM left hereditary. All rings considered are assumed to be associative. The concept "class of rings" shall mean "isomorphically closed class of rings". The expression 0 L <~t R (0 ~ L <~ R) will denote that L is a non-zero left (two-sided) ideal