Abstract :
LetRbe a domain andKits quotient-field. For a subsetSofK, let imageR(S) be the set of polynomialsfset membership, variantK[x] withf(S)subset of or equal toRand define theR-closure ofSas the set of thosetset membership, variantKfor whichf(t)set membership, variantRfor allfset membership, variantimageR(S). The concept ofR-closure was introduced by McQuillan (J. Number Theory39(1991), 245–250), who gave a description in terms of closure inP-adic topology, whenRis a Dedekind ring with finite residue fields. We introduce a toplogy related to, but weaker thanP-adic topology, which allows us to treat ideals of infinite index, and derive a characterization ofR-closure whenRis a Krull ring. This gives us a criterion for imageR(S)=imageR(T), whereSandTare subsets ofK. As a corollary we get a generalization to Krull rings ofR. Gilmerʹs result (J. Number Theory33(1989), 95–100) characterizing those subsetsSof a Dedekind ring with finite residue fields for which imageR(S)=imageR(R).
