LetSbe a finite direct sum of Dedekind prime rings. We shall study those subringsRofSsuch thatRcontains a suitable idealBofSThe aim is to derive general properties ofRfrom those ofS. For instance we show that every ideal ofRwhich is comaximal withBis invertible. We also show that ifRis right Noetherian thenRhas right Krull dimension 1 and every right ideal ofRwhich is comaximal withBis projective, andRhas the right Artin-Rees property if and only if the maximal ideals ofRwhich containBcommute with each other.