On the base radical class for associative rings

The base radical class Lb (X) generated by a class X was introduced in [12]. It consists of those rings whose nonzero homomorphicimages have nonzero accessible subrings in X. When X is homomorphically closed, Lb(X) is the lower radical class defined by X, but otherwise X may not be contained in Lb(X). We prove that for a hereditary radical class R with semisimple class S(R), Lb(S(R)) is the class of strongly R-semisimple rings if and only if R is supernilpotent or subidempotent. A number of further examples of radical classes of the form Lb(X) are discussed.