Radical Ideals in Valuation Domains

Abstract

AbstractAn idealIof a ringRis called a radical ideal ifI= ℛ(R) where ℛ is a radical in the sense of Kurosh–Amitsur. The main theorem of this paper asserts that ifRis a valuation domain, then a proper idealIofRis a radical ideal if and only ifIis a distinguished ideal ofR(the latter property means that ifJandKare ideals ofRsuch thatJ⊂I⊂Kthen we cannot haveI/J≅K/Ias rings) and that such an ideal is necessarily prime. Examples are exhibited which show that, unlike prime ideals, distinguished ideals are not characterizable in terms of a property of the underlying value group of the valuation domain.