Computing the strong metric dimension for co-maximal ideal graphs of commutative rings

Metric dimension in a prime ideal sum graph of a commutative ring

The prime ideal sum graph of a commutative unital ring $R$, denoted by $PIS(R)$, is an undirect and simple graph whose vertices are non-trivial ideals of $R$ and there exists and edge between to distinct vertices if and only if their sum is a prime ideal of $R$. In this paper, the metric dimension of $PIS(R)$ is discussed and some formulae for this parameter in $PIS(R)$ are given.

A Note on Co-Maximal Ideal Graph of Commutative Rings

Let $R$ be a commutative ring with unity. The co-maximal ideal graph of $R$, denoted by $Γ(R)$, is a graph whose vertices are the proper ideals of $R$ which are not contained in the Jacobson radical of $R$, and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1 + I_2 = R$. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012 the following question was posed: If $Γ(R)$ is an infinite star graph, can $R$ be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question.

交换环的素理想和图的强度量维数

Strong metric dimension of the prime ideal sum graph of a commutative ring

本文研究含单位元交换环的素理想和图，并确定了多类 Artin 非局部交换环对应图的强度量维数。

自由交换幺半群的理想扩张

Ideal extensions of free commutative monoids

本文引入间隙吸收幺半群，证明其链次度至多四且长度集为区间，并给出理想扩张的ω-素性界及有限间隙的等价条件。

Some Properties of the Nil-Graphs of Ideals of Commutative Rings

Let $R$ be a commutative ring with identity and ${\rm Nil}(R)$ be the set of nilpotent elements of $R$. The nil-graph of ideals of $R$ is defined as the graph $\mathbb{AG}_N(R)$ whose vertex set is $\{I:\ (0)\neq I\lhd R$ and there exists a non-trivial ideal $J$ such that $IJ\subseteq {\rm Nil}(R)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ\subseteq {\rm Nil}(R)$. Here, we study conditions under which $\mathbb{AG}_N(R)$ is complete or bipartite. Also, the independence number of $\mathbb{AG}_N(R)$ is determined, where $R$ is a reduced ring. Finally, we classify Arti…

Decompositions of commutative monoid congruences and binomial ideals

Primary decomposition of commutative monoid congruences is insensitive to certain features of primary decomposition in commutative rings. These features are captured by the more refined theory of mesoprimary decomposition of congruences, introduced here complete with witnesses and associated prime objects. The combinatorial theory of mesoprimary decomposition lifts to arbitrary binomial ideals in monoid algebras. The resulting binomial mesoprimary decomposition is a new type of intersection decomposition for binomial ideals that enjoys computational efficiency and independence from ground fiel…

On graphs related to co-maximal ideals of a commutative ring

This paper studies the co-maximal graph $\Om(R)$, the induced subgraph $\G(R)$ of $\Om(R)$ whose vertex set is $R\setminus (U(R)\cup J(R))$ and a retract $\G_r(R)$ of $\G(R)$, where $R$ is a commutative ring. We show that the core of $\G(R)$ is a union of triangles and rectangles, while a vertex in $\G(R)$ is either an end vertex or a vertex in the core. For a non-local ring $R$, we prove that both the chromatic number and clique number of $\G(R)$ are identical with the number of maximal ideals of $R$. A graph $\G_r(R)$ is also introduced on the vertex set $\{Rx|\,x\in R\setminus (U(R)\cup J(R…

More on the Annihilator-Ideal Graph of a Commutative Ring

Let $R$ be a commutative ring with identity and $\Bbb A (R)$ be the set of ideals of $R$ with non-zero annihilator. The annihilator-ideal graph of $R$, denoted by $A_{I} (R) $, is a simple graph with the vertex set $\Bbb A(R)^{\ast} := \Bbb A (R) \setminus\lbrace (0) \rbrace $, and two distinct vertices $I$ and $J$ are adjacent if and only if $\mathrm{Ann} _{R} (IJ) \neq \mathrm{Ann} _{R} (I) \cup \mathrm{Ann} _{R} (J)$. In this paper, we study the affinity between the annihilator-ideal graph and the annihilating-ideal graph $\Bbb A \Bbb G (R)$ (a well-known graph with the same vertices and tw…

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.

关于素理想和图为线图的环

On rings whose prime ideal sum graphs are line graphs

该研究针对含单位元的交换环，围绕素理想和图的结构性质展开，刻画了素理想和图为线图及线图补图的所有交换Artinian环，丰富了环与图交叉领域的结构刻画结果。

具有平面、环面或射影平面素理想和图的环的刻画

Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs

该文研究含单位元交换环的素理想和图的图论性质与环代数性质的关联，分类了交叉帽数至多2、亏格为1的非局部交换Artin环，证明不存在素理想和图为射影平面的这类环。

图与超图上的持久交换代数

Persistent commutative algebra on graphs and hypergraphs

本文提出持久交换代数用于研究图与超图边理想在过滤下的代数与组合演变，通过持久Hochster公式和Betti分裂不等式，揭示了持久边理想在数据科学中的应用价值。

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…