Stefania Gabelli, Evan Houston, Thomas G. Lucas
We study a condition on intersections of localizations of a domain at maximal t-ideals. This extends and generalizes earlier work of Gilmer (1967), Gilmer-Heinzer (1968), Olberding (1998), and others for Prufer domains.
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To study the question of whether every two-dimensional Prüfer domain possesses the stacked bases property, we consider the particular case of the Prüfer domains formed by integer-valued polynomials. The description of the spectrum of the rings of integer-valued polynomials on a subset of a rank-one valuation domain enables us to prove that they all possess the stacked bases property. We also consider integer-valued polynomials on rings of integers of number fields and we reduce in this case the study of the stacked bases property to questions concerning $2\times 2$-matrices.
An integral domain is said to have the IDF property when every non-zero element of it has only a finite number of non-associate irreducible divisors. A counterexample has already been found showing that IDF property does not necessarily ascend in polynomial extensions. In this paper, we introduce a new class of integral domains, called MCD-finite domains, and show that for any domain $D$, $D[X]$ is an IDF domain if and only if $D$ is both IDF and MCD-finite. This result entails all the previously known sufficient conditions for the ascent of the IDF property. Our new characterization of polyno…
An integral domain $D$ is called an irreducible-divisor-finite domain (IDF-domain) if every nonzero element of $D$ has finitely many irreducible divisors up to associates. The study of IDF-domains dates back to the seventies. In this paper, we investigate various aspects of the IDF property. In 2009, P.~Malcolmson and F. Okoh proved that the IDF property does not ascend from integral domains to their corresponding polynomial rings, answering a question posed by D. D. Anderson, D. F. Anderson, and the second author two decades before. Here we prove that the IDF property ascends in the class of …
On Integral Domains with Prime Divisor Finite Property
本文研究了紧密具有素因子有限性质的整环，探讨了其基本性质及在局部化、$D+M$构造和多项式环中的行为。
The complete integral closure of a Prüfer domain is a topological property
该研究证明了Prüfer整环的完全整闭包的素谱完全由其自身素谱上的Zariski拓扑决定，揭示了整环代数性质与拓扑结构的关联。
On GL-domains and the ascent of the IDF property
本文研究GL-整环类，基于素性类元素给出其刻画并识别新子类，还证明GL-整环类中IDF性质不可提升至多项式扩张，回答了相关开放问题。
We extend to torsion-free modules over integral domains the theory of (pre)-Schreier domains initiated by Cohn and Zafrullah.
In this article we study two classes of integral domains. The first is characterized by having a finite intersection of principal ideals being finitely generated only when it is principal. The second class consists of the integral domains in which a finite intersection of principal ideals is always non-finitely generated except in the case of containment of one of the principal ideals in all the others. We relate these classes to many well-studied classes of integral domains, to star operations and to classical and new ring constructions.
Let $R$ be an integral domain. For elements $a,b \in R$, let $[a,b]$ denote their greatest common divisor, if it exists. We say that $R$ has the Z-property if whenever $a,b,c,d$ and $e$ are nonzero nonunits of $R$ such that $abc=de$, then $[ab,d] \neq 1$ or $[ab,e] \neq 1$. The purpose of this paper is to study this property. The atomic integral domains that have this property constitute a class of half-factorial domains. Also, it is known that $R$ must have this property in order for the polynomial ring $R[x]$ to be half-factorial. We use it to give a characterization of half-factorial polyno…
Let $D$ be an integral domain. A nonzero nonunit $a$ of $D$ is called a valuation element if there is a valuation overring $V$ of $D$ such that $aV\cap D=aD$. We say that $D$ is a valuation factorization domain (VFD) if each nonzero nonunit of $D$ can be written as a finite product of valuation elements. In this paper, we study some ring-theoretic properties of VFDs. Among other things, we show that (i) a VFD $D$ is Schreier, and hence ${\rm Cl}_t(D)=\{0\}$, (ii) if $D$ is a P$v$MD, then $D$ is a VFD if and only if $D$ is a weakly Matlis GCD-domain, if and only if $D[X]$, the polynomial ring o…
An unrestricted notion of the finite factorization property
本文引入并研究了有限分解性质的无限制版本（U-FF），证明了其与IDF性质的类同性，并在IDF与FF条件间提供更精细的插值。
The Regular property of Invariant Rings over Regular Domains
本文研究了伪反射群在正则域上不变环的正则性，证明了群由伪反射生成当且仅当不变环为正则环。
Let R be a commutative ring and I an ideal of R. A sub-ideal J of I is a reduction of I if JI^n = I^n+1 for some positive integer n. The ring R has the (finite) basic ideal property if (finitely generated) ideals of R do not have proper reductions. Hays characterized (one-dimensional) Prufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays' results to Prufer v-multiplication domains by replacing "basic" with "w-basic," where w is a particular star operation. We also investigate relations among star-basic properties for certain star operations.
On near atomicity and a characterization of the FF property
本文构造了一个非原子的近原子整环，并证明整环是FFD当且仅当它是近原子的且具有IDF性质，且该刻画中近原子性不能弱化为几乎原子性。
Let $G$ be a group acting via ring automorphisms on an integral domain $R.$ A ring-theoretic property of $R$ is said to be $G$-invariant, if $R^G$ also has the property, where $R^G=\{r\in R \ | \ σ(r)=r \ \text{for all} \ σ\in G\},$ the fixed ring of the action. In this paper we prove the following classes of rings are invariant under the operation $R\rightarrow R^G:$ locally pqr domains, Strong G-domains, G-domains, Hilbert rings, $S$-strong rings and root-closed domains. Further let $\mathscr{P}$ be a ring theoretic property and $R\subseteq S$ be a ring extension. A pair of rings $(R,S)$ is …
Local properties of integral domains under extensions and pullback constructions
本文研究了积分域在扩展和拉回构造下局部性质的传递，探讨了平坦上环、Nagata理想变换、多项式环及其商扩展等构造对局部性质的影响。
On the axioms for a unique factorization domain
针对整环和幺半群中非唯一分解理论的发展，研究经典唯一分解整环（UFD）性质的变体，利用长度分解性质证明UFD定义中部分经典公理多余。
We extend to Prüfer $v$-multiplication domains some distinguished ring-theoretic properties of Prüfer domains. In particular we consider the $t##$-property, the $t$-radical trace property, $w$-divisoriality and $w$-stability.
Let $R$ be an integral domain with $qf(R)=K$ and let $F(R)$ be the set of nonzero fractional ideals of $R.$ Call $R$ a dually compact domain (DCD) if for each $I\in F(R)$ the ideal $I_{v}=(I^{-1})^{-1}$ is a finite intersection of principal fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori and Krull domains. In addition we show that a Schreier DCD is a GCD domain with the property that for each $A\in F(R)$ the ideal $A_{v}$ is principal. We show that a domain $R$ is G-Dedekind domain (i.e. has…
On graded going-down domains, II
本文研究了回拉中分级整环的分级下行性质，并据此构造了该类整环的原创性实例。
Given a certain factorization property of a ring $R$, we can ask if this property extends to the polynomial ring over $R$ or vice versa. For example, it is well known that $R$ is a unique factorization domain if and only if $R[X]$ is a unique factorization domain. If $R$ is not a domain, this is no longer true. In this paper we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.
This paper seeks ring-theoretic conditions of an integral domain R that reflect in the Clifford property or Boolean property of its class semigroup S(R), that is, the semigroup of the isomorphy classes of the nonzero (integral) ideals of R with the operation induced by multiplication. Precisely, in Section 3, we characterize integrally closed domains with Boolean class semigoup; in this case, S(R) identifies with the Boolean semigroup formed of all fractional overrings of R. In Section 4, we investigate Noetherian-like settings where the Clifford and Boolean properties of S(R) coincide with (L…
On the ideal avoidance property
本文研究了理想与环的回避性质，提出了回避引理的一般版本，证明了幂等理想、回避整环的过度环等具有回避性，并刻画了直积与平坦环满态射下的保持性，同时引入强回避概念。
Recently, many authors have embraced the study of certain properties of modules such as projectivity, injectivity and flatness from an alternative point of view. Rather than saying a module has a certain property or not, each module is assigned a relative domain which, somehow, measures to which extent it has this particular property. In this work, we introduce a new and fresh perspective on flatness of modules. However, we will first investigate a more general context by introducing domains relative to a precovering class $\x$. We call these domains $\x$-precover completing domains. In partic…
Factorizations into idempotent factors of matrices over Prüfer domains
针对刻画满足幂等矩阵分解性质的整环这一经典问题，研究证明满足(ID2)的整环必为满足特定可逆矩阵分解条件的Prüfer域，并验证了一大类平面曲线坐标环与Int(Z)符合相关猜想的等价形式。
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