General Heredity and Strength for Radical Classes

Abstract

An H-relation, as introduced by Rossa and Tangeman [4], is a relation σ on the class of associative rings with their subrings satisfying the following conditions:(1) IσR implies that I is a subring of R;(2) if IσR and f is a homomorphism of R, then (If)σ(Rf);(3) if IσR and J is an ideal of R, then (I∩J)σJ.Puczylowski [3] imposes also the condition(4) if J is an ideal of R, then JσR.A further condition satisfied by many familiar H-relations is the following:(5) if f is a homomorphism from a ring R onto a ring S and BσS, then there exists a subring A of R such that AσR and Af = B.