Genera of zero-divisor graphs with respect to ideals

Asymptotic Prime Divisors Related to Ext, Regularity of Powers of Ideals, and Syzygy Modules

This dissertation focuses on the following topics: (1) asymptotic prime divisors over complete intersection rings, (2) asymptotic stability of complexities over complete intersection rings, (3) asymptotic linear bounds of Castelnuovo-Mumford regularity in multigraded modules, and (4) characterizations of regular local rings via syzygy modules of the residue field.

支配二维正规诺特局部整环的素除子的交的理想理论

The ideal theory of intersections of prime divisors dominating a normal Noetherian local domain of dimension two

本文研究二维正规诺特局部整环上特定水平下素除子的交的理想理论，证明该交为几乎Dedekind整环且非零理想可唯一表示为极大理想幂的无赘交。

关于交换环的基于理想的余零因子图的一些结果

Some results on the ideal-based cozero-divisor graph of a commutative ring

本文研究含单位元交换环R关于理想I的余零因子图Γ''_I(R)，给出了该图的一些性质结果。

A commutative Bezout domain in which every maximal ideal is principal is an elementary divisor ring

In this article we revisit a problem regarding Bezout domains, namely, whether every Bezout domain is an elementary divisor domain. We prove that a Bezout domain in which every maximal ideal is principal is an elementary divisor ring

i-扩展的理想基余零除数图

The $i$-extended ideal-based cozero-divisor graph of a commutative ring

本文研究了交换环的i-扩展理想基余零除数图，定义了图的顶点和边的条件，并探讨了其结构性质。

A little more on the zero-divisor graph and the annihilating-ideal graph of a reduced ring

We have tried to translate some graph properties of AG(R) and Gamma(R) to the topological properties of Zariski topology. We prove that Rad(Gamma(R)) and Rad(AG(R)) are equal and they are equal to 3, if and only if the zero ideal of R is an anti fixed-place ideal, if and only if Min(R) does not have any isolated point, if and only if Gamma(R) is triangulated, if and only if AG(R) is triangulated. Also, we show that if the zero ideal of a ring R is a fixed-place ideal, then dtt(AG(R)) = |B(R)| and also if in addition |Min(R)| > 2, then dt(AG(R)) = |B(R)|. Finally, it has been shown that dt(AG(R…

具有零因子的交换环中理想化的若干分解性质

Some factorization properties of idealization in commutative rings with zero divisors

本文研究交换环中模的理想化的分解性质，证明 ACCP 的等价条件，说明 BF 性质不一定保持并给出保持条件，同时刻画了理想化环为 UFR 的情形。

Primary decomposition of the ideal of polynomials whose fixed divisor is divisible by a prime power

We characterize the fixed divisor of a polynomial $f(X)$ in $\mathbb{Z}[X]$ by looking at the contraction of the powers of the maximal ideals of the overring ${\rm Int}(\mathbb{Z})$ containing $f(X)$. Given a prime $p$ and a positive integer $n$, we also obtain a complete description of the ideal of polynomials in $\mathbb{Z}[X]$ whose fixed divisor is divisible by $p^n$ in terms of its primary components.

与根系相关的环面环及通过拟阵理论的圆锥除子理想

Toric rings associated with root systems and conic divisorial ideals via matroid theory

从拟阵理论视角研究圆锥除子理想，并将其应用于根系环面环，通过拟阵描述除子类对应的多面体，计算符号偏序集P对应的环面环R_P的除子类群并刻画其Q-Gorenstein性质。

Divisors of a module and blow up

In this paper we work with several divisors of a module $E \subseteq G \simeq R^{e}$ having rank $e$, such as the classical Fitting ideals of $E$ and of $G/E$, and the more recently introduced (generic) Bourbaki ideals $I(E)$ (A. Simis, B. Ulrich, and W. Vasconcelos [18]) or ideal norms $[[E]]_R$ (O. Villamayor [22]). We found several relations and equalities among them which allow to describe in some cases universal properties with respect to $E$ of their blow ups. As a byproduct we are also able to obtain lower bounds for the analytic spread $\ell(\bigwedge^{e}E)$ related with the algebraic …

Multi-Rees algebras on principal ideal rings

When $R$ is a Noetherian ring and we have a family of ideals in which every ideal contains at least one nonzero divisor, then it is already known that the defining ideal of the multi-Rees algebra of these ideals is equal to a saturated ideal. In such a case to get the defining ideal of the multi-Rees algebra we only need to saturate the first syzygies of direct sum of this family of ideals. However, this fact is not true, when at least one of these ideals does not contain any nonzero divisor. In this paper we show that the defining ideal of the multi-Rees algebra of a family of ideals of a pol…

On Modules of Finite Projective Dimension

We address two aspects of finitely generated modules of finite projective dimension over local rings and their connection in between: embeddability and grade of order ideals of minimal generators of syzygies. We provide a solution of the embeddability problem and prove important reductions and special cases of the order ideal conjecture. In particular we derive that in any local ring R of mixed characteristic p > 0, where p is a non-zero-divisor, if I is an ideal of finite projective dimension over R and p is in I or p is a non-zero-divisor on R/I, then every minimal generator of I is a non-ze…

Divisorial linear algebra of normal semigroup rings

We investigate the minimal number of generators $μ$ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound $C$ there exist, up to isomorphism, only finitely divisorial ideals $I$ such that $μ(I)\le C$. It follows that there exist only finitely many Cohen--Macaulay divisor classes. Moreover we determine the minimal depth of all divisorial ideals and the behaviour of $μ$ and depth in ``arithmetic progressions'' in t…

On reduced zero-divisor graphs of posets

In this paper we study some of the basic properties of a graph which is constructed from the equivalence classes of non-zero zero-divisors determined by annihilator ideals of a poset. In particular, we demonstrate how this graph helps in identifying the annihilator prime ideals of a poset that satisfies the ascending chain condition for its proper annihilator ideals.

环面环的锥除子理想及其在Hibi环和稳定集环中的应用

Conic divisorial ideals of toric rings and applications to Hibi rings and stable set rings

本文研究环面环的锥除子理想，提出确定方法并描述了完美图的Hibi环与稳定集环的锥除子理想，刻画了其对称性，并构造了特殊稳定集环族的NCCR。

Conic divisorial ideals of Hibi rings and their applications to non-commutative crepant resolutions

In this paper, we study divisorial ideals of a Hibi ring which is a toric ring arising from a partially ordered set. We especially characterize the special class of divisorial ideals called conic using the associated partially ordered set. Using our description of conic divisorial ideals, we also construct a module giving a non-commutative crepant resolution (= NCCR) of the Segre product of polynomial rings. Furthermore, applying the operation called mutation, we give other modules giving NCCRs of it.

符号解析扩张与除子解析扩张是有限的

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。

Cochordal 零因子图及其边理想的 Betti 数

Cochordal zero divisor graphs and Betti numbers of their edge ideals

本文通过引入类型序列计算 cochordal 图边理想的分级 Betti 数，并分类了使 $\mathbb{Z}/n \mathbb{Z}$ 零因子图为 cochordal 的正整数 $n$，确定了其边理想的 Betti 数。

Coloring of zero-divisor graphs of posets and applications to graphs associated with algebraic structures

In this paper, we characterize chordal and perfect zero-divisor graphs of finite posets. Also, it is proved that the zero-divisor graphs of finite posets and the complement of zero-divisor graphs of finite $0$-distributive posets satisfy the Total Coloring Conjecture. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graph of rings, the annihilating ideal graphs, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of cyclic groups. In fact, it is proved that these graphs associated with a commutative ring $R$ with…

Conic divisor classes over a normal monoid algebra

Let R=K[M] be a normal affine monoid algbera over a field K.Up to isomorphism the conic ideals are exactly the direct summands ofthe extension R^{1/n} of R. We show that the classes of the conic divisorial ideals can be identified with the full-dimensional open cells in a decomposition of a torus naturally associated with M. Furthermore, they can be characterized by the relative compactness of a certain group associated with them. Baetica has given examples of Cohen-Macaulay divisorial ideals that are not conic. Wereview his construction and streamline the arguments somewhat. In the last part …

满足零因子性质的Hurwitz级数环

Hurwitz series rings satisfying a zero divisor property

本文研究非交换环上Hurwitz级数环的零因子，构造并分析了多种Armendariz环，证明了半素环在多项式扩张下保持Hurwitz级数的Armendariz性质。

交换环的基于理想的拟非零因子图

Ideal-based quasi cozero divisor graph of a commutative ring

本文在有单位元的交换环中引入并研究了一个基于理想的拟非零因子无向图$QΓ''_I(R)$，通过结合根理想$\sqrt{I}$与理想I的条件重新定义了图的顶点集与邻接关系。

Zero-Divisors of Content Algebras

In this article, we prove that in content extentions minimal primes extend to minimal primes and discuss zero-divisors of a content algebra over a ring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also examine the preservation of diameter of zero-divisor graph under content extensions.

Divisors on graphs, binomial and monomial ideals, and cellular resolutions

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their $\mathbb{Z}$-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other r…

A finiteness property of graded sequences of ideals

Given a graded sequence of ideals (a_m) on a smooth variety $X$ having finite log canonical threshold, suppose that for every m we have a divisor E_m over X that computes the log canonical threshold of a_m, and such that the log discrepancies of the divisors E_m are bounded. We show that in this case the set of divisors E_m is finite.

Divisors on graphs, orientations, syzygies, and system reliability

We study various ideals arising in the theory of system reliability. We use ideas from the theory of divisors, orientations and matroids on graphs to describe the minimal polyhedral cellular free resolutions of these ideals. In each case we give an explicit combinatorial description of the minimal generating set for each higher syzygy module in terms of the acyclic orientations of the graph, the $q$-reduced divisors and the bounded regions of the graphic hyperplane arrangement. The resolutions of all these ideals are closely related, and their Betti numbers are independent of the characteristi…

Invertibility of ideals in Prufer extensions

Using the general approach to invertibility for ideals in ring extensions given by Knebush-Zhang, we investigate about connections between faithfully flatness and invertibility for ideals in rings with zero divisors.

Conic divisorial ideals and non-commutative crepant resolutions of edge rings of complete multipartite graphs

The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by $\Bbbk[K_{r_1,\ldots,r_n}]$, where $1 \leq r_1 \leq \cdots \leq r_n$. More concretely, we prove that the class group of $\Bbbk[K_{r_1,\ldots,r_n}]$ is isomorphic to $\mathbb{Z}^n$ if $n =3$ with $r_1 \geq 2$ or $n \geq 4$, while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of $\Bbbk[K_{r_1,\ldots,r_n}]$, called conic divisorial ideals. We describe conic divisorial ideals fo…

Hodge理想的一种算法

An algorithm for Hodge ideals

本文提出一种计算与任意有效除子关联的Hodge理想算法，基于Malgrange和Kashiwara的V-滤波算法，并提供了新的计算乘子理想和跳跃数的方法。

A canonical linear system associated to adjoint divisors in characteristic $p > 0$

Suppose that $X$ is a projective variety over an algebraically closed field of characteristic $p > 0$. Further suppose that $L$ is an ample (or more generally in some sense positive) divisor. We study a natural linear system in $|K_X + L|$. We further generalize this to incorporate a boundary divisor $Δ$. We show that these subsystems behave like the global sections associated to multiplier ideals, $H^0(X, \mJ(X, Δ) \tensor L)$ in characteristic zero. In particular, we show that these systems are in many cases base-point-free. While the original proof utilized Kawamata-Viehweg vanishing and va…

Quasi-finite modules and asymptotic prime divisors

Let $A$ be a Noetherian ring, $J\subseteq A$ an ideal and $C$ a finitely generated $A$-module. In this note we would like to prove the following statement. Let $\{I_n\}_{n\geq 0}$ be a collection of ideals satisfying : (i) $I_n\supseteq J^n$, for all $n$, (ii) $J^s\cdot I_s \subseteq I_{r+s}$, for all $r,s\geq 0$ and (iii) $I_n\subseteq I_m$, whenever $m\leq n$. Then $\Ass_A(I_nC/J^nC)$ is independent of $n$, for $n$ sufficiently large. Note that the set of prime ideals $\cup_{n\geq 1} \Ass_A(I_nC/J^nC)$ is finite, so the issue at hand is the realization that the primes in $\Ass_A(I_nC/J^nC)$ …

Analytic spread of filtrations on two dimensional normal local rings

In this paper we prove that a classical theorem by McAdam about the analytic spread of an ideal in a Noetherian local ring continues to be true for divisorial filtrations on a two dimensional normal excellent local ring $R$, and that the Hilbert polynomial of the fiber cone of a divisorial filtration on $R$ has a Hilbert function which is the sum of a linear polynomial and a bounded function. We prove these theorems by first studying asymptotic properties of divisors on a resolution of singularities of the spectrum of $R$. The filtration of the symbolic powers of an ideal is an example of a di…

Prûfer property in amalgamated algebras along an ideal

Let $f : A \rightarrow B$ be a ring homomorphism and $J$ be an ideal of $B$. In this paper, we give a characterization of zero divisors of the amalgamation which is a generalization of Maimani's and Yassemi's work (see \cite{y}). Also, we investigate the transfer of Prüfer domain concept to commutative rings with zero divisors in the amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by $A\bowtie^fJ),$ introduced and studied by D'Anna, Finocchiaro and Fontana in 2009 (see \cite{AFF1} and \cite{AFF2}). Our aim is to provide new classes of commutative rings satisfying this prope…

环沿理想的融合复制的扩展零因子图

The extended zero-divisor graph of the amalgamated duplication of a ring along an ideal

本文研究环沿理想的融合复制的扩展零因子图，刻画其与零因子图重合及完全性的条件，并计算直径和围长。

通过除子体积的超凡ε重数

Transcendental Epsilon Multiplicity via Divisor Volumes

本文证明ε重数可以取超越数，通过一个单理想公式将渐近余长度不变量转化为除子体积的几何与算术，并利用移位射影丛构造和Baker定理得到超越值。

Ideals generated by diagonal 2-minors

With a simple graph $G$ on $[n]$, we associate a binomial ideal $P_G$ generated by diagonal minors of an $n \times n$ matrix $X=(x_{ij})$ of variables. We show that for any graph $G$, $P_G$ is a prime complete intersection ideal and determine the divisor class group of $K[X]/ P_G$. By using these ideals, one may find a normal domain with free divisor class group of any given rank.

On freeness of divisors on $\mathbb P^2$

Let $I\subset \mathbb C[x,y,z]$ be an ideal of height 2 and minimally generated by three homogeneous polynomials of the same degree. If $I$ is a locally complete intersection we give a criterion for $\mathbb C[x,y,z]/I$ to be arithmetically Cohen-Macaulay. Since the setup above is most commonly used when $I=J_F$ is the Jacobian ideal of the defining polynomial of a "quasihomogeneous" reduced curve $Y=V(F)$ in $\mathbb P^2$, our main result becomes a criterion for freeness of such divisors. As an application we give an upper bound for the degree of the reduced Jacobian scheme when $Y$ is a free…

Non-commutative Krull monoids: A divisor theoretic approach and their arithmetic

A (not necessarily commutative) Krull monoid---as introduced by Wauters---is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these results we give a criterion for a Krull monoid to be a bounded factorization monoid, and we provide arithmetical finiteness results in case of normalizing Krull monoids with finite Davenport constant.

环中一般线性形式是精确零因子

Rings for which general linear forms are exact zero divisors

本文研究了特征为零的域上标准格雷德环中一般线性形式为精确零因子的性质，并提出了关于此类环Hilbert函数的猜想，证明了在特定条件下该猜想成立。

On zero-divisors of semimodules and semialgebras

In Section 1 of the paper, we prove McCoy's property for the zero-divisors of polynomials in semirings. We also investigate zero-divisors of semimodules and prove that under suitable conditions, the monoid semimodule $M[G]$ has very few zero-divisors if and only if the $S$-semimodule $M$ does so. The concept of Auslander semimodules are introduced in this section as well. In Section 2, we introduce Ohm-Rush and McCoy semialgebras and prove some interesting results for prime ideals of monoid semirings. In Section 3, we investigate the set of zero-divisors of McCoy semialgebras. We also introduc…

An interpretation of multiplier ideals via tight closure

Hara and Smith independently proved that in a normal $\mQ$-Gorenstein ring of characteristic $p \gg 0$, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair $(R, Δ)$ of a normal ring $R$ and an effective $\mQ$-Weil divisor $Δ$ on $\Spec R$. As a corollary, we obtain the equivalence of strongly F-regular pairs and klt pairs.

A Note on Bernstein-Sato Varieties for Tame Divisors and Arrangements

For strongly Euler-homogeneous, Saito-holonomic, and tame analytic germs we consider general types of multivariate Bernstein-Sato ideals associated to arbitrary factorizations of our germ. We show the zero loci of these ideals are purely codimension one and the zero loci associated to different factorizations are related by a diagonal property. If, additionally, the divisor is a hyperplane arrangement, we show the Bernstein-Sato ideals attached to a factorization into linear forms are principal. As an application, we independently verify and improve an estimate of Maisonobe's regarding standar…

Divisors on graphs, Connected flags, and Syzygies

We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner bases for each higher syzygy module. In each case the given minimal Gröbner bases is also a minimal generating set. The Betti numbers of the binomial ideal and its natural initial ideal coincide and they correspond to the number of 'connected flags' in the graph. In particular the Betti numbers are independent of the characteristic of the base field. For complete graphs the problem was previously studied b…

Content Algebras Over Commutative Rings With Zero-Divisors

Let $M$ be an $R$-module and $c$ the function from $M$ to the ideals of $R$ defined by $c(x) = \cap \lbrace I \colon I \text{is an ideal of} R \text{and} x \in IM \rbrace $. $M$ is said to be a content $R$-module if $x \in c(x)M $, for all $x \in M$. $B$ is called a content $R$-algebra, if it is a faithfully flat and content $R$-module and it satisfies the Dedekind-Mertens content formula. In this article, we prove some new results for content modules and algebras by using ideal theoretic methods.

Multiplier ideals in two-dimensional local rings with rational singularities

The aim of this paper is to study jumping numbers and multiplier ideals of any ideal in a two-dimensional local ring with a rational singularity. In particular we reveal which information encoded in a multiplier ideal determines the next jumping number. This leads to an algorithm to compute sequentially the jumping numbers and the whole chain of multiplier ideals in any desired range. As a consequence of our method we develop the notion of jumping divisor that allows to describe the jump between two consecutive multiplier ideals. In particular we find a unique minimal jumping divisor that is s…

关于带零因子Prüfer环的一些注记

Some remarks on Prüfer rings with zero-divisors

本文通过纤维积构造研究了带零因子Prüfer环的Prüfer类性质刻画，给出了新构造及同态像保持Prüfer性的例子，并讨论了pre-Prüfer环的理想理论。

Matching numbers and the regularity of the Rees algebra of an edge ideal

The regularity of the Rees ring of the edge ideal of a finite simple graph is studied. We show that the matching number is a lower and matching number~$+1$ is an upper bound of the regularity, if the Rees algebra is normal. In general the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.

Prufer conditions in an amalgamated duplication of a ring along an ideal

This paper investigates ideal-theoretic as well as homological extensions of the Prufer domain concept to commutative rings with zero divisors in an amalgamated duplication of a ring along an ideal. The new results both compare and contrast with recent results on trivial ring extensions (and pullbacks) as well as yield original families of examples issued from amalgamated duplications subject to various Prufer conditions.

On realizing zero-divisor graphs of po-semirings

In this paper, we determine bipartite graphs and complete graphs with horns, which are realizable as zero-divisor graphs of po-semirings. As applications, we classify commutative rings $R$ whose annihilating-ideal graph $\mathbb {AG}(R)$ are either bipartite graphs or complete graphs with horns.

Divisors on Rational Normal Scrolls

Let $A$ be the homogeneous coordinate ring of a rational normal scroll. The ring $A$ is equal to the quotient of a polynomial ring $S$ by the ideal generated by the two by two minors of a scroll matrix $ψ$ with two rows and $\ell$ catalecticant blocks. The class group of $A$ is cyclic, and is infinite provided $\ell$ is at least two. One generator of the class group is $[J]$, where $J$ is the ideal of $A$ generated by the entries of the first column of $ψ$. The positive powers of $J$ are well-understood, in the sense that the $n^{\text{th}}$ ordinary power, the $n^{th}$ symmetric power, and th…

Prüfer- like conditions in the amalgamated duplication of a ring along an ideal

In this paper, we consider five possible extensions of the Prüfer domain notion to the case of commutative rings with zero-divisors. We investigate the transfer of these Prüfer-like properties between a ring $R$ and $R\bowtie I$; his amalgamated duplication along some ideal $I$, and then generate new and original families of rings with these properties.

融合代数的零因子图

The zero-divisor graph of an amalgamated algebra

本文研究融合代数零因子图的直径计算问题，推广并改进已有结果，给出零因子图完备性的新刻画，并完整描述有限环情形下的直径。

Zero-divisor graphs of nilpotent-free semigroups

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an \emph{Armendariz map} between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtai…

DG 模结构与模精确零因子的极小自由分解

DG module structures and minimal free resolutions modulo an exact zero-divisor

该研究针对满足fM=0的有限Q-模，证明其在Q上的极小自由分解具有特定微分分次代数上的DG模结构，并借助该结构刻画了模精确零因子时的极小自由分解。