On the Base Radical Class for Associative Rings

The base radical class â„’b(X), generated by a class X was introduced in [12]. It consists of those rings whose nonzero homomorphic images have nonzero accessible subrings in X. When X is homomorphically closed, â„’b(X) is the lower radical class defined by X, but otherwise X may not be contained in â„’b(X). We prove that for a hereditary radical class R with semisimple class S(R), â„’b (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 â„’b(X) are discussed.