Said El Baghdadi, Stefania Gabelli
We study the class of domains in which each w-ideal is divisorial, extending several properties of divisorial and totally divisorial domains to a much wider class of domains. In particular we consider PvMDs and Mori domains.
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Alternating Divisoriality of Prime Ideals along Infinite Chains of Bézout Domains
针对 Prüfer 域超环链上素理想可除性是否可无限交替的问题，构造所有有限阶段为 Bézout 域的链，证明极限素理想可除性在升降两个方向均不确定。
It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this paper we show that a domain without {\em prime} strongly divisorial ideals is not necessarily completely integrally closed, although this property holds under some additional assumptions.
We extend the Bass-Matlis characterization of local Noetherian divisorial domains to the non-Noetherian case. This result is then used to study the following question: If a domain D is w-divisorial, that is, if each w-ideal of D is divisorial, then is D[X] automatically w-divisorial? We show that the answer is yes if D is either integrally closed or Mori.
Divisorial Multiplicative Lattices
该研究在乘法格的理论框架下，针对除子整环展开研究，证明了相关的若干基本结论，丰富了乘法格领域的研究成果。
We give conditions for a maximal divisorial ideal to be t-maximal and show with examples that, even in a completely integrally closed domain, maximal divisorial ideals need not be t-maximal.
Let $D$ be a Prüfer $\star$-multiplication domain, where $\star$ is a semistar operation on $D$. We show that certain ideal-theoretic properties related to idempotence and divisoriality hold in Prüfer domains, and we use the associated semistar Nagata ring of $D$ to show that the natural counterparts of these properties also hold in $D$.
The symbolic and divisorial analytic spreads are finite
该研究证明域上本质有限型环的理想符号幂最小生成元个数为多项式有界，整环的ℝ-除子滤过理想的最小生成元个数也呈多项式有界，明确符号与除子解析扩张有限性。
Residual functions and divisorial ideals
该研究提出拓扑空间上剩余函数的定义，并将其用于研究Prüfer整环上除子理想群的自由性，为相关代数结构研究提供新工具。
Suppose that $R$ is an excellent local domain with maximal ideal $m_R$. The theory of multiplicities and mixed multiplicities of $m_R$-primary ideals extends to (possibly non Noetherian) filtrations of $R$ by $m_R$-primary ideals, and many of the classical theorems for $m_R$-primary ideals continue to hold for filtrations. The celebrated theorems involving inequalities continue to hold for filtrations, but the good conclusions that hold in the case of equality for $m_R$-primary ideals do not hold for filtrations. In this article, we consider multiplicities and mixed multiplicities of $R$ by $m…
Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of $v$-invertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions we establish arithmetical finiteness results, in particular for the monotone catenary degree and for the structure of sets of lengths and of their unions.
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.
A commutative ring $R$ is stable if every non-zero ideal $I$ of $R$ is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the $2$-generator property. In the present paper we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains.
The Rees algebra and analytic spread of a divisorial filtration
本文研究除子滤链的Rees代数与解析展度，将McAdam定理推广到$\mathbb Q$-除子滤链，证明其一阶差分函数的上极限有限但极限可能不存在，并构造了Rees赋值集合无限的符号滤链例子。
We show that in certain Prüfer domains, each nonzero ideal $I$ can be factored as $I=I^v Π$, where $I^v$ is the divisorial closure of $I$ and $Π$ is a product of maximal ideals. This is always possible when the Prüfer domain is $h$-local, and in this case such factorizations have certain uniqueness properties. This leads to new characterizations of the $h$-local property in Prüfer domains. We also explore consequences of these factorizations and give illustrative examples.
The primary purpose of this paper is give a classification scheme for the nonzero primes of a Prüfer domain based on five properties. A prime $P$ of a Prüfer domain $R$ could be sharp or not sharp, antesharp or not, divisorial or not, branched or unbranched, idempotent or not. Based on these five basic properties, there are six types of maximal ideals and twelve types of nonmaximal (nonzero) primes. Both characterizations and examples are given for each type that exists.
C-domains are defined via class semigroups, and every C-domain is a Mori domain with nonzero conductor whose complete integral closure is a Krull domain with finite class group. In order to extend the concept of C-domains to rings with zero divisors, we introduce $v$-Marot rings as generalizations of ordinary Marot rings and study their theory of regular divisorial ideals. Based on this we establish a generalization of a result well-known for integral domains. Let $R$ be a $v$-Marot Mori ring, $\hat R$ its complete integral closure, and suppose that the conductor $\mathfrak f = (R : \hat R)$ i…
et $R$ be an integral domain with quotient field $L$. An overring $T$ of $R$ is $t$-linked over $R$ if $I^{-1}=R$ implies that $(T:IT)=T$ for each finitely generated ideal $I$ of $R$. Let $O_{t}(R)$ denotes the set of all $t$-linked overrings of $R$ and $O(R)$ the set of all overrings of $R$. The purpose of this paper is to study some finiteness conditions on the set $O_{t}(R)$. Particularly, we prove that if $O_{t}(R)$ is finite, then so is $O(R)$ and $O_{t}(R)=O(R)$, and if each chain of $t$-linked overrings of $R$ is finite, then each chain of overrings of $R$ is finite. This yields that th…
Gorenstein Prüfer domains with non-Prüfer integral closure
该研究证明存在一维局部Gorenstein Prüfer整环，其整闭包非Prüfer，还确立了有限生成理想除子性在超积与可分素理想商下的不变性，提供一致矩阵分解并区分相关失败情形。
Primary and strongly primary monoids and domains play a central role in the ideal and factorization theory of commutative monoids and domains. It is well-known that primary monoids satisfying the ascending chain condition on divisorial ideals (e.g., numerical monoids) are strongly primary; and the multiplicative monoid of non-zero elements of a one-dimensional local domain is primary and it is strongly primary if the domain is Noetherian. In the present paper, we focus on the study of additive submonoids of the non-negative rationals, called Puiseux monoids. It is easy to see that Puiseux mono…
We consider the lattice-ordered groups Inv$(R)$ and Div$(R)$ of invertible and divisorial fractional ideals of a completely integrally closed Prüfer domain. We prove that Div$(R)$ is the completion of the group Inv$(R)$, and we show there is a faithfully flat extension $S$ of $R$ such that $S$ is a completely integrally closed Bézout domain with Div$(R) \cong $ Inv$(S)$. Among the class of completely integrally closed Prüfer domains, we focus on the one-dimensional Prüfer domains. This class includes Dedekind domains, the latter being the one-dimensional Prüfer domains whose maximal ideals are…
Extending valuations of local domains to complete local domains without changing the value group
研究在不改变值群的情况下，将局部域的赋值扩展到完备局部域的问题，通过给出泰西耶猜想的证明，为正特征下局部一致化的两种方法提供重要步骤。
Let $V$ be a valuation domain with quotient field $K$. We show how to describe all extensions of $V$ to $K(X)$ when the $V$-adic completion $\widehat{K}$ is algebraically closed, generalizing a similar result obtained by Ostrowski in the case of one-dimensional valuation domains. This is accomplished by realizing such extensions by means of pseudo-monotone sequences, a generalization of pseudo-convergent sequences introduced by Chabert. We also show that the valuation rings associated to pseudo-convergent and pseudo-divergent sequences (two classes of pseudo-monotone sequences) roughly corresp…
Given an arbitrary graph G, we study its basic covers algebra, which is the symbolic fiber cone of the Alexander dual of the edge ideal of G. Extending results of Villarreal and Benedetti-Constantinescu-Varbaro, valid only in the case when G is bipartite, we characterize in a combinatorial fashion the situations when: 1) the basic covers algebra is a domain, and 2) it is a domain and in addition (the edge ideal of) G is unmixed. It turns out that the last result gives a complete characterization of those graphs for which any symbolic power of the edge ideal is generated by monomials of the sam…
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