Radically perfect prime ideals in polynomial rings

We call an ideal I of a commutative ring R radically perfect if among the ideals of R whose radical is equal to the radical of I the one with the least number of generators has this number of generators equal to the height of I. Let R be a Noetherian integral domain of Krull dimension one containing a field of characteristic zero. Then each prime ideal of the polynomial ring R[X] is radically perfect if and only if R is a Dedekind domain with torsion ideal class group. We also show that over a finite dimensional Bézout domain R, the polynomial ring R[X] has the property that each prime ideal of it is radically perfect if and only if R is of dimension one and each prime ideal of R is the radical of a principal ideal.     

Radically Perfect Prime Ideals in Polynomial Rings Over Prüfer and Pullback Rings

In this article, we study the notion of radical perfectness in Prüfer and classical pullbacks issued from valuation domains. We answer positively a question by Erdogdu of whether a domain R such that every prime ideal of the polynomial ring R[X] is radically perfect is one-dimensional. Particularly, we prove that Prüfer and pseudo-valuation domains R over which every prime ideal of the polynomial ring R[X] is radically perfect are one-dimensional domains. Moreover, the class group of such a Prüfer domain is torsion.     

Pullbacks of Arithmetical Rings

We give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (i.e., a ring whose ideals are totally ordered by inclusion). We also give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (i.e., a ring which is locally a chain ring at every maximal ideal). For any integral domain D with field of fractions K, we characterize all Prüfer domains R between D[X] and K[X] such that the conductor C of K[X] into R is nonzero. As an application, we show that for n ≥ 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.     

Maximally Prüfer Rings

In this paper we consider six Prüfer-like conditions on a commutative ring R, and introduce seventh condition by defining the ring R to be maximally Prüfer if R _M is Prüfer for every maximal ideal M of R, and we show that the class of such rings lie properly between Prüfer rings and locally Prüfer rings. We give a characterization of such rings in terms of the total quotient ring and the core of the regular maximal ideals. We also find a relationship of such rings with strong Prüfer rings.     

Minimal Overrings of an Integrally Closed Domain

Abstract

The main purpose of this paper is to characterize minimal overrings of an integrally closed domain R. We show that there exists a strong relationship between minimal overrings and the notion of ideal transforms. In particular, we prove that if T(M) = S(M) for each maximal ideal M, then there is a bijective correspondence between the set of invertible maximal ideals of R and the set of minimal overrings of R. This study enables us to produce several interesting applications concerning semi-local, Dedekind, Prüferian and Krull domains. Moreover, we investigate the spectrum of a minimal overring in comparison with the spectrum of R, and we determine whether the polynomial ring R[X ₁, X ₂,…, X _n ] has a minimal overring.     

Umt-domains and domains with prüfer integral closure

An integral domain R is said to be a UMT-domain if uppers to zero in R[X) are maximal t-ideals. We show that R is a UMT-domain if and only if its localizations at maximal tdeals have Prüfer integral closure. We also prove that the UMT-property is preserved upon passage to polynomial rings. Finally, we characterize the UMT-property in certian pullback constructions; as an application, we show that a domain has Prüfer integral closure if and only if all its overrings are UMT-domains.     

Integrally Closed Domains with Only Finitely Many Star Operations

We prove that an integrally closed domain R admits only finitely many star operations if and only if R satisfies each of the following conditions: (1) R is a Prüfer domain with finite character, (2) all but finitely many maximal ideals of R are divisorial, (3) only finitely many maximal ideals of R contain a nonzero prime ideal that is contained in some other maximal ideal of R, and (4) if P ≠ (0) is the largest prime ideal contained in a (necessarily finite) collection of maximal ideals of R, then the prime spectrum of R/P is finite.     

Uppers to Zero in Polynomial Rings and Prüfer-Like Domains

Let D be an integral domain and X an indeterminate over D. It is well known that (a) D is quasi-Prüfer (i.e., its integral closure is a Prüfer domain) if and only if each upper to zero Q in D[X] contains a polynomial g ∈ D[X] with content c _D (g) = D; (b) an upper to zero Q in D[X] is a maximal t-ideal if and only if Q contains a nonzero polynomial g ∈ D[X] with c _D (g) ^v  = D. Using these facts, the notions of UMt-domain (i.e., an integral domain such that each upper to zero is a maximal t-ideal) and quasi-Prüfer domain can be naturally extended to the semistar operation setting and studied in a unified frame. In this article, given a semistar operation ☆ in the sense of Okabe–Matsuda, we introduce the ☆-quasi-Prüfer domains. We give several characterizations of these domains and we investigate their relations with the UMt-domains and the Prüfer v-multiplication domains.     

Note on Star Operations Over Polynomial Rings

This article studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in R[X] is a *-maximal ideal and when a *-maximal ideal Q of R[X] is extended from R, that is, Q = (Q ∩ R)[X] with Q ∩ R ≠0, for a given star operation of finite character * on R[X]. We also answer negatively some questions raised by Anderson–Clarke by constructing a Prüfer domain R for which the v-operation is not stable.     

PrÜfer-Like Domains and the Nagata Ring of Integral Domains

A subring A of a Prüfer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A?B is a unibranched extension and (ii) there exists a nonzero radical ideal I, common to A and B such that each prime ideal of A (resp., B) containing I is maximal in A (resp., B). Let D be an integral domain, X be an indeterminate over D, c(f) be the ideal of D generated by the coefficients of a polynomial f ∈ D[X], N = {f ∈ D[X] | c(f) = D}, and N _v  = {f ∈ D[X] | c(f)^?1 = D}. In this article, we study when the Nagata ring D[X] _N (more generally, D[X] _{N v} ) is a GPVD. To do this, we first use the so-called t-operation to introduce the notion of t-globalized pseudo-valuation domains (t-GPVDs). We then prove that D[X] _{N v} is a GPVD if and only if D is a t-GPVD and D[X] _{N v} has Prüfer integral closure, if and only if D[X] is a t-GPVD, if and only if each overring of D[X] _{N v} is a GPVD. As a corollary, we have that D[X] _N is a GPVD if and only if D is a GPVD and D has Prüfer integral closure. We also give several examples of integral domains D such that D[X] _{N v} is a GPVD.     

Strong S-domains

S-domains and strong S-rings are studied extensively with special emphasis on integral and polynomial ring extensions. The main theorem of this paper is that for a Prüfer domain R, the polynomial ring R[X₁,…X_n] in finitely many indeterminates is a strong S-domain. We also prove that any Prüfer υ-multiplication domain is an S-domain.     

Krull Dimension in Power Series Ring Over an Almost Pseudo-Valuation Domain

Let R be an integral domain. We say that R is a star-domain if R has at least a height one prime ideal and if for each height one prime ideal P of R, R satisfies the acc on P-principal ideals (i.e., ideals of the form aP, a ∈ R). We prove that if R is an APVD with nonzero finite Krull dimension, then the power series ring R[[X]] has finite Krull dimension if and only if R is a residually star-domain (i.e., for each nonmaximal prime ideal P of R, R/P is a star-domain) if and only if R[[X]] is catenarian.     

Localization of Injective Modules Over Arithmetical Rings

It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P, R _P is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreover, if R is a Prüfer domain of finite character, localizations of injective R-modules are injective.     

Integer-Valued Polynomial Rings,t-Closure,and Associated Primes

Given an integral domain D with quotient field K, the ring of integer-valued polynomials on D is the subring {f(X) ∈ K[X]: f(D) ? D} of the polynomial ring K[X]. Using the tools of t-closure and associated primes, we generalize some known results on integer-valued polynomial rings over Krull domains, Prüfer v-multiplication domains, and Mori domains.     

Polynomials Contained in a Finite Number of Maximal Ideals

Abstract

Let R[X] be a polynomial ring in one variable over a commutative ring R. If (R,?) is a local ring then any Weierstrass polynomial in R[X] is contained only in the maximal ideal (?,X) of R[X]. We generalise this property of Weierstrass polynomials and investigate properties of polynomials contained in a finite number of maximal ideals in R[X].     

The Gröbner ring conjecture in one variable

We prove that a valuation domain V has Krull dimension ≤ 1 if and only if for every finitely generated ideal I of V[X] the ideal generated by the leading terms of elements of I is also finitely generated. This proves the Gröbner ring conjecture in one variable. The proof we give is both simple and constructive. The same result is valid for semihereditary rings.     

Semistar-Krull and valuative dimension of integral domains

Given a stable semistar operation of finite type ⋆ on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite type ⋆[X] on the polynomial ring D[X], such that, if n := ⋆-dim(D), then n+1 ≤ ⋆[X]-dim(D[X]) ≤ 2n+1. We also establish that if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain, then ⋆[X]-dim(D[X]) = ⋆- dim(D)+1. Moreover we define the semistar valuative dimension of the domain D, denoted by ⋆-dim _v (D), to be the maximal rank of the ⋆-valuation overrings of D. We show that ⋆-dim _v (D) = n if and only if ⋆[X ₁, . . . , X _n ]-dim(D[X ₁, . . . , X _n ]) = 2n, and that if ⋆-dim _v (D) < ∞ then ⋆[X]-dim _v (D[X]) = ⋆-dim _v (D) + 1. In general ⋆-dim(D) ≤ ⋆-dim _v (D) and equality holds if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain. We define the ⋆-Jaffard domains as domains D such that ⋆-dim(D) < ∞ and ⋆-dim(D) = ⋆-dim _v (D). As an application, ⋆-quasi-Prüfer domains are characterized as domains D such that each (⋆, ⋆′)-linked overring T of D, is a ⋆′-Jaffard domain, where ⋆′ is a stable semistar operation of finite type on T. As a consequence of this result we obtain that a Krull domain D, must be a w _D -Jaffard domain.