An H-relation, as introduced by Rossa and Tangeman , 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  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.