Idempotent ideals and unions of nets of Prüfer domains

Abstract

In this paper, all rings considered are assumed to be commutative rings with an identity element. It is known that an integral domain D may contain an idempotent proper ideal A. But when this occurs, A is not finitely generated [21, p. 215], so that D is not Noetherian. Also, it is easy to show that for any positive integer k there exists a ring R which is not a domain and such that R contains an ideal A with the property that A^)A^) •-•^)A = A=. .. Whether an integral domain R with this property exists is a heretofore open question which we answer affirmatively in §2. Nakano in [16] has considered the problem of determining when an ideal of D is idempotent, where D is the integral closure of Z, the domain of ordinary integers, in an infinite algebraic number field. In fact, the paper [16] is one of a series of papers which Nakano has written concerning the ideal structure of D. In [18], Ohm has generalized and simplified many of Nakano's results from [16] and [17], showing that as far as the structure of the set of primary ideals of D is concerned, the assumption that D is the integral closure of Z in an algebraic number field is superfluous the essential requirement on D being that it is a Prύfer domain according to the following definition: The integral domain / is a Prϋfer domain if for each proper prime ideal P of /, JP is a valuation ring; equivalently, / is a Prϋfer domain if each nonzero finitely generated ideal of / is invertible [10, p. 554]. Following Ohm's example, we show in §3 that most of Nakano's results in [16] carry over to the case when D is the integral closure of a fixed Prϋfer domain Do in an algebraic extension of the quotient field of Do. If / is an integral domain with quotient field K, a domain /0 between / and K will be called an overrίng of /. In case /0 is a valuation ring, we call /o a valuation overring of /. We say that / is an almost Dedekind domain if for each maximal ideal M of /, JM is a rank one discrete valuation ring [5], in