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.     

A Characterization of Prüfer Domains

In this article, it is proved that a domain R is a Prüfer domain if and only if it is coherent, integrally closed and FP-id _R (R) ≤ 1.     

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.     

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.     

Towards a classification of stable semistar operations on a Prüfer domain

We study stable semistar operations defined over a Prüfer domain, showing that, if every ideal of a Prüfer domain R has only finitely many minimal primes, every such closure can be described through semistar operations defined on valuation overrings of R.     

The set of indeterminate rings of a normal pair as a partially ordered set

Let R ì S{R\subset S} be an extension of integral domains and let [R, S] be the set of intermediate rings between R and S ordered by inclusion. If (R, S) is normal pair and [R, S] is finite, we do prove that there exists a semi-local Prüfer ring T with quotient field K such that [R,S] @ [T,K]{[R,S]\cong \lbrack T,K]} (as a partially ordered set). Consequently, any problem relative to the finiteness conditions in [R, S] can be investigated in the particular case where R is a semi-local Prüfer ring with quotient field S.     

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.     

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.     

On sums and products of primitive elements

Let R be a (commutative integral) domain, with K its quotient field and R^′ its integral closure (in K). Let 𝒫 be the set of elements u∈K such that u is primitive over R; i.e., such that u is the root of a polynomial over R having a unit coe?cient. Then, 𝒫 is a ring (necessarily K) ? 𝒫 is closed under products ? R^′ is a Prüfer domain. In general, 𝒫 is closed under powers. For u,v∈𝒫, necessary and su?cient conditions are given for u+v (resp., uv) to belong to 𝒫. Also, 𝒫 is used to characterize when R is a quasi-local integrally closed domain and when R is a pseudo-valuation domain. If R is quasi-local, each element of K is expressible as the sum of two (possibly equal) elements of 𝒫. The set of primitive elements is determined for lying-over pairs and for extensions of domains with the same sets of prime ideals. In this study of the 𝒫 construction, R and K are replaced, whenever possible, by an arbitrary commutative ring and its total quotient ring or, more generally, by any inclusion of commutative rings.     

Integral Domains Which Admit at Most Two Star Operations

In this article, we characterize domains which admit at most two star operations in the integrally closed and Noetherian cases. We also precisely count the number of star operations on an h-local Prüfer domain.     

Butler Modules Over Prüfer Domains

If R is an integral domain, let  be the class of torsion free completely decomposable R-modules of finite rank. Denote by  the class of those torsion-free R-modules A such that A is a homomorphic image of some C ? , and let 𝒫 be the class of R-modules K such that K is a pure submodule of some C ? . Further, let Q  and Q 𝒫 be the respective closures of  and 𝒫 under quasi-isomorphism. In this article, it is shown that if R is a Prüfer domain, then Q  = Q 𝒫, and  = 𝒫 in the special case when R is h-local. Also, if R is an h-local Prüfer domain and if C ?  has a linearly ordered typeset, it is established that all pure submodules and all torsion-free homomorphic images of C are themselves completely decomposable. Finally, as an application of these results, we prove that if R is an h-local Prüfer domain, then  = Q  = Q 𝒫 = 𝒫 if and only if R is almost maximal.     

Ring extensions with some finiteness conditions on the set of intermediate rings

A ring extension R ⊆ S is said to be FO if it has only finitely many intermediate rings. R ⊆ S is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension R ⊆ S to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair with only finitely many intermediate rings. We also obtain as a corollary several new and old characterizations of Prüfer and integral domains satisfying the corresponding finiteness conditions.     

On Extensions of Semilocal Prüfer Domains

One of the most important results of Chevalley's extension theorem states that every valuation domain has at least one extension to every extension field of its quotient field. We state a generalization of this result for Prüfer domains with any finite number of maximal ideals. Then we investigate extensions of semilocal Prüfer domains in algebraic field extensions. In particular, we find an upper bound for the cardinality of extensions of a semilocal Prüfer domain. Moreover, we show that any two extensions of a semilocal Prüfer domain are incomparable (by inclusion) in an algebraic extension of fields.     

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.     

On Finite Maximal Chains of Weak Baer Going-Down Rings

Some recent results of Ayache on going-down domains and extensions of domains that either are residually algebraic or have DCC on intermediate rings are generalized to the context of extensions of commutative rings. Given a finite maximal chain 𝒞 of R-subalgebras of a weak Baer ring T, it is shown how a “min morphism” hypothesis can be used to transfer the “going-down ring” property from R to each member of 𝒞. The integral minimal ring extensions which are min morphisms are classified. The ring extensions satisfying FCP (i.e., for which each chain of intermediate rings is finite) are characterized as the strongly affine extensions with DCC on intermediate rings. In the relatively integrally closed case, such extensions R ? T induce open immersions Spec(S) → Spec(R) for each R-subalgebra S of T.     

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.     

Injective Modules over Prüfer v-Multiplication Domains

Let R be an integral domain with quotient field F. It is shown that R is a strongly discrete Prüfer v-multiplication domain if and only if there exists a bijection between the set of the prime w-ideals and the set of isomorphism classes of GV-torsionfree indecomposable injective R-modules and every indecomposable injective R-module, viewed as a module over its endomorphism ring, is uniserial. It is also shown that the w-closure of any GV-torsionfree homomorphic image of F is injective if and only if R is a Prüfer v-multiplication domain satisfying an almost maximality-type property.