Ira J. Papick

When the Dual of an Ideal is a Ring.

Let R be a commutative integral domain with identity with quotient field K, and let I be a nonzero ideal of R. We analyze several general and particular instances when I−1 is a subring of K. We then apply some of our results to show that certain non-maximal prime ideals in Prüfer domains are divisorial.

The radical trace property

or II-’ E Spec(R) [4, Prop. 2.41. Using this formulation of TP, Fontana, Huckaba and Papick gave characterizations for Noetherian TP domains and a special class of Priifer TP domains [4]. It is our goal to extend this work by studying RTP (radical trace property) domains, i.e., domains satisfying the condition that ZZ- ’ =

M-Canonical ideals in integral domains

We prove that a Priifer domain R has an m-canonical ideal J, that is, an ideal I such that J: (I: J) = J for every ideal J of R, if and only if R is h-local with only finitely many maximal ideals that are not finitely generated; moreover, if these conditions are satisfied, then the product of the non-finitely generated maximal ideals is an m-canonical ideal of R

When is

Let V be a valuation ring of the form K + M, where K is a field and M(# 0) is the maximal ideal of V. Let D be a proper subring of K. Necessary and sufficient conditions are given that the ring D + M be coherent. The condition that a given ideal of V be D + A/-flat is also characterized.

D+M

E H om,(M, R)}. When no confusion will result, we will suppress the R and write y(M). In Section 2, we prove that if R is a valuation domain and M a unital R-module, then y(M) either equals R or a prime ideal of R. It is this result that motivates our work and gives rise to the following definition: A domain R is said to satisfy the

coherent?

Let P be a prime ideal of a Prtifer domain R. In [3] we studied when P is a divisorial ideal; i.e., when (P‘)’ = P. If P is a maximal ideal, then it is known that P is divisorial if and only if P is finitely generated, [lo, Corollary 3.41. When P is a non-maximal prime ideal, we gave several sufficient conditions for P to be divisorial, [3]. However, the characterization of non-maximal divisorial prime ideals was left open. In Section 2 of this paper, we estabish in Proposition 9 the desired characterization of divisorial prime ideals. Also, in Proposition 7, we give equivalent conditions for a non-idempotent prime ideal with the property that P-’ = T(P) to be divisorial, and in Theorem 8, we characterize those prime ideals for which each power is divisorial. In Cpptinn 3 annlir9tinnc 9re oiven fnr 9 cnmGol rlarc nf Priifnr Anrn~~;ne ln mar a.. “1ILl”‘l J, UppllrULlVllCl UlU b..bU 1”I u 0prru.a s.IUJO “A 1 1 U1b1 U”l‘lQlllD. 1‘1 patitular, for Priifer domains for which each overring satisfies (#), (see [6]), it is proved that the product of divisorial prime ideals is divisorial. Finally, examples are given to show that each prime ideal of a Priifer domain may be divisorial, yet not all ideals of the ring are divisorial. That is, there is no Cohen type theorem for divisorial ideals in Prufer domains. The following nntntinn will he find thrnllrrhnllt thic naner 1 et R he a Priifm rln D __________ . . ___ __ _A.___ .._a__O”__’ --a*” y...y”a. YIC a. “1 u * -“awn uvmain with quotient field K, and let Spec(R) denote the set of prime ideals of R. If PE Spec(R), let {A&) denote the set of maximal ideals of R that do not contain P. Define S=(n R,)nK. The ideal transform of P is T(P) = lJr=, (RK: P”). When no ambiguity may arise, write (R : I) instead of (RK: I). For the prime ideal P,

Domains satisfying the trace property

result, we will write c(f),S, U, and (a:b). It follows that both 5 and U are multiplicatively closed sets in R[x] [7, Proposition 33.1], [17, Theorem F], and that R[x]s Q R[x]n. The ring R[x]s, denoted by R(x), has been the object of study of several authors (see for example [1], [2], [3], [12]). An especially interesting paper concerning R(x) is that of Arnolds [3], where he, among other things, characterizes when R(x) is a Priifer domain. We shall make special use of his results in our work. In § 2 we determine conditions on the ring R so that R(x) = R[x]v. A complete characterization of this property is given for Noetherian domains in Proposition 2.2. In particular, we prove that if R is a Noetherian domain, then R(x) = R[x]v if and only if depth (R) ^ 1. Some sufficient conditions for R(x) = R[x]u are that R be treed (Proposition 2.5), or that SP (R) (see § 2 for définitions) be finite (Proposition 2.9). The main results of this paper occur in § 3. We prove that if R is either a GCD-domain, an integrally closed coherent domain, or a Krull domain, then R[x]u is a Bezout domain. As is well known [7, Theorem 32.7], the Kronecker function ring R of an integrally closed domain R is a Bezout domain. Hence, it would seem likely that R and R[x]v would coincide for many rings R. However, this is not the case. In fact, R = R[x]v if and only if R is a Priifer domain. In general there is no containment relation between R and R[x]v (Remark 3.3). Finally, we apply the results of § 3 to § 1 to obtain new characterizations of Priifer domains, Bezout domains, and Dedekind domains (Corollary 3.2).

Some properties of divisorial prime ideals in Prüfer domains

Let R be a commutative ring with identity. The multiplicatively closed sets U2={f∈R[X]: c(f)−1=R}, (U2)={f∈U2: f is regular} and S={f∈R[X]: c(f)=R} are studied. By considering various equalities between these sets, many characterizations of Noetherian rings are found. In particular, a Noetherian ring R has depth ≤1 if and only if S=(U2): and each maximal ideal of a Noetherian ring is regular if and only if U2=(U2).The theory of Prüfer v-multiplication rings (PVMRs) is developed for rings with zero divisors. Six equivalent conditions are given to the statement that an additively regular v-ring R is a PVMR.

A localization of

Given a Priifer domain R and a prime ideal P in R, we study some conditions which force P to be a divisorial ideal of R. This paper extends some recent work of Huckaba and Papick.

R[x]

An extensionR⊆T of commutative integral domains is called a Δ0-extension, provided each intermediateR-module is actually an intermediate ring, and an extensionR⊆T is called quadratic if eacht∈T satisfies a monic quadratic polynomial overR. Our purpose is to investigate these extensions in the context of Prüfer domains.