Comaximal Ideal Graph Research Articles

Eulerian and pancyclic zero-divisor graphs of ordered sets

In this paper, we determine when the zero-divisor graph of a special class of a finite pseudocomplemented poset is Eulerian. Also, we deal with Hamiltonian, vertex pancyclic, and edge pancyclic properties of the complement of a zero-divisor graph of finite 0-distributive posets. These results are applied to study the Eulerian, Hamiltonian and pancyclic properties of the zero-divisor graph, comaximal ideal graph, annihilating ideal graph, intersection ideal graph of a special class of commutative rings and intersection graph of a special class of groups.

Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures

ABSTRACT In this paper, we characterize the perfect zero-divisor graphs and chordal zero-divisor graphs (its complement) of ordered sets. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graphs of rings, the annihilating ideal graphs of rings, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of groups. In fact, it is shown that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R.

On graphs with equal coprime index and clique number

Recently, Katre et al. introduced the concept of the coprime index of a graph. They asked to characterize the graphs for which the coprime index is the same as the clique number. In this paper, we partially solve this problem. In fact, we prove that the clique number and the coprime index of a zero-divisor graph of an ordered set and the zero-divisor graph of a ring Z p n coincide. Also, it is proved that the annihilating ideal graphs, the co-annihilating ideal graphs and the comaximal ideal graphs of commutative rings can be realized as the zero-divisor graphs of specially constructed posets. Hence the coprime index and the clique number coincide for these graphs as well.

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

Let [Formula: see text] be a commutative ring with identity. The co-maximal ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph whose vertices are proper ideals of [Formula: see text] which are not contained in the Jacobson radical [Formula: see text] of [Formula: see text] and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we use Gallai’s Theorem and the concept of strong resolving graph to compute the strong metric dimension for co-maximal ideal graphs of commutative rings. Explicit formulae for the strong metric dimension, depending on whether the ring is reduced or not, are established.

Structural Properties of the Essential Ideal Graph of <img src=image/13424240_01.gif>

Let <img src=image/13424240_02.gif> be a commutative ring with unity. The essential ideal graph of <img src=image/13424240_02.gif>, denoted by <img src=image/13424240_03.gif>, is a graph with vertex set consisting of all nonzero proper ideals of A and two vertices <img src=image/13424240_04.gif> and <img src=image/13424240_05.gif> are adjacent whenever <img src=image/13424240_06.gif> is an essential ideal. An essential ideal <img src=image/13424240_04.gif> of a ring <img src=image/13424240_02.gif> is an ideal <img src=image/13424240_04.gif> of <img src=image/13424240_02.gif> (<img src=image/13424240_07.gif>), having nonzero intersection with every other ideal of <img src=image/13424240_02.gif>. The set <img src=image/13424240_08.gif> contains all the maximal ideals of <img src=image/13424240_02.gif>. The Jacobson radical of <img src=image/13424240_02.gif>, <img src=image/13424240_09.gif>, is the set of intersection of all maximal ideals of <img src=image/13424240_02.gif>. The comaximal ideal graph of <img src=image/13424240_02.gif>, denoted by <img src=image/13424240_11.gif>, is a simple graph with vertices as proper ideals of A not contained in <img src=image/13424240_09.gif> and the vertices <img src=image/13424240_04.gif> and <img src=image/13424240_05.gif> are associated with an edge whenever <img src=image/13424240_10.gif>. In this paper, we study the structural properties of the graph <img src=image/13424240_03.gif> by using the ring theoretic concepts. We obtain a characterization for <img src=image/13424240_03.gif> to be isomorphic to the comaximal ideal graph <img src=image/13424240_11.gif>. Moreover, we derive the structure theorem of <img src=image/13424240_12.gif> and determine graph parameters like clique number, chromatic number and independence number. Also, we characterize the perfectness of <img src=image/13424240_12.gif> and determine the values of <img src=image/13424240_13.gif> for which <img src=image/13424240_12.gif> is split and claw-free, Eulerian and Hamiltonian. In addition, we show that the finite essential ideal graph of any non-local ring is isomorphic to <img src=image/13424240_12.gif> for some <img src=image/13424240_13.gif>.

A note on comaximal ideal graph of commutative rings

Let R be a commutative ring with identity. The comaximal ideal graphG(R) of R is a simple graph with its vertices are the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1+I2=R. In this paper, a dominating set of G(R) is constructed using elements of the center when R is a commutative Artinian ring. Also we prove that the domination number of G(R) is equal to the number of factors in the Artinian decomposition of R. Also, we characterize all commutative Artinian rings(non local rings) with identity for which G(R) is planar.

Some results on the comaximal ideal graph of a commutative ring

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring such that R admits at least two maximal ideals.

Some results on the comaximal ideal graph of a commutative ring

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let $R$ be a ring such that $R$ admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of $R$, denoted by $\mathscr{C}(R)$ is an undirected simple graph whose vertex set is the set of all proper ideals $I$ of $R$ such that $I\not\subseteq J(R)$, where $J(R)$ is the Jacobson radical of $R$ and distinct vertices $I_{1}$, $I_{2}$ are joined by an edge in $\mathscr{C}(R)$ if and only if $I_{1} + I_{2} = R$. In Section 2 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is planar. In Section 3 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is a split graph. In Section 4 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is complemented and moreover, we determine the $S$-vertices of $\mathscr{C}(R)$.

Some results on the complement of the comaximal ideal graphs of commutative rings

The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let R be a ring. Recall from Ye and Wu (J Algebra Appl 11(6):1250114, 2012) that the comaximal ideal graph of R denoted by $${\mathscr {C}}(R)$$ is an undirected graph whose vertex set is the set of all proper ideals I of R such that $$I\not \subseteq J(R)$$ , where J(R) is the Jacobson radical of R and distinct vertices I, J are joined by an edge in this graph if and only if $$I + J = R$$ . The aim of this article is to study the interplay between the ring-theoretic properties of R and the graph-theoretic properties of $$({\mathscr {C}}(R))^{c}$$ , where $$({\mathscr {C}}(R))^{c}$$ is the complement of the comaximal ideal graph of R.

Automorphisms of the co-maximal ideal graph over matrix ring

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] be the ring of all [Formula: see text] matrices over [Formula: see text], [Formula: see text] be the set of all nontrivial left ideals of [Formula: see text]. The co-maximal ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph with [Formula: see text] as vertex set and two nontrivial left ideals [Formula: see text] of [Formula: see text] are adjacent if and only if [Formula: see text]. If [Formula: see text], it is easy to see that [Formula: see text] is a complete graph, thus any permutation of vertices of [Formula: see text] is an automorphism of [Formula: see text]. A natural problem is: How about the automorphisms of [Formula: see text] when [Formula: see text]. In this paper, we aim to solve this problem. When [Formula: see text], a mapping [Formula: see text] on [Formula: see text] is proved to be an automorphism of [Formula: see text] if and only if there is an invertible matrix [Formula: see text] and a field automorphism [Formula: see text] of [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text].

A note on a graph related to the comaximal ideal graph of a commutative ring

‎The rings considered in this article are commutative with identity which admit at least two maximal ideals‎. ‎This article is inspired by the work done on the comaximal ideal graph of a commutative ring‎. ‎Let R be a ring‎. ‎We associate an undirected graph to R denoted by mathcal{G}(R)‎, ‎whose vertex set is the set of all proper ideals I of R such that Inotsubseteq J(R)‎, ‎where J(R) is the Jacobson radical of R and distinct vertices I1‎, ‎I2are adjacent in mathcal{G}(R) if and only if I1∩ I2 = I1I2‎. ‎The aim of this article is to study the interplay between the graph-theoretic properties of mathcal{G}(R) and the ring-theoretic properties of R.

Some properties of comaximal ideal graph of a commutative ring

Let $R$ be a commutative ring with identity‎. ‎We use‎ ‎$varphi (R)$ to denote the comaximal ideal graph‎. ‎The vertices‎ ‎of $varphi (R)$ are proper ideals of R which are not contained‎ ‎in the Jacobson radical of $R$‎, ‎and two vertices $I$ and $J$ are‎ ‎adjacent if and only if $I‎ + ‎J = R$‎. ‎In this paper we show some‎ ‎properties of this graph together with planarity of line graph‎ ‎associated to $varphi (R)$‎.

Some results on the comaximal ideal graph of a commutative ring

Let $R$ be a commutative ring with unity. The comaximal ideal graph of $R$, denoted by $mathcal{C}(R)$, 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 $I_1 +I_2 = R$. In this paper, we classify all comaximal ideal graphs with finite independence number and present a formula to calculate this number. Also, the domination number of $mathcal{C}(R)$ for a ring $R$ is determined. In the last section, we introduce all planar and toroidal comaximal ideal graphs. Moreover, the commutative rings with isomorphic comaximal ideal graphs are characterized. In particular we show that every finite comaximal ideal graph is isomorphic to some $mathcal{C}(mathbb{Z}_n)$.

Co-maximal ideal graphs of matrix algebras

Let R be a ring with unity. The co-maximal ideal graph of R, denoted by \(\Gamma (R)\), is a graph whose vertices are all non-trivial left ideals of R, and two distinct vertices \(I_1\) and \(I_2\) are adjacent if and only if \(I_1 + I_2 = R\). In this paper, some results on the co-maximal ideal graphs of matrix algebras are given. For instance, we determine the domination number, the clique number and a lower bound of the independence number of \(\Gamma (M_n(\mathbb {F}_q))\), where \(M_n(\mathbb {F}_q)\) is the ring of \(n\times n\) matrices over the finite field \(\mathbb {F}_q\). Furthermore, we characterize all rings (not necessarily commutative) whose domination numbers of their co-maximal ideal graphs are finite. Among other results, we show that if \(\Gamma (R)\cong \Gamma (M_n(\mathbb {F}_q))\), where \(n\ge 2\) is a positive integer and R is a ring, then \(R\cong M_n(\mathbb {F}_q)\). Also, it is proved that if R and \(R'\) are two finite reduced rings and \(\Gamma (M_m(R))\cong \Gamma (M_n(R'))\), for some positive integers \(m,n\ge 2\), then \(m=n\) and \(R\cong R'\).

On the comaximal ideal graph of a commutative ring

Let $R$ be a commutative ring with identity. We use $\Gamma ( R )$ to denote the comaximal ideal graph. The vertices of $\Gamma ( R )$ are proper ideals of R that are not contained in the Jacobson radical of $R$, and two vertices $I$ and $J$ are adjacent if and only if $I + J = R$. In this paper we show some properties of this graph together with the planarity and perfection of $\Gamma ( R )$.

Complemented graphs and blow-ups of Boolean graphs, with applications to co-maximal ideal graphs

For a set X, let 2X be the power set of X. Let BX be the Boolean graph, which is defined on the vertex set 2X \ {X, ?}, with M adjacent to N if M ? N = ?. In this paper, several purely graph-theoretic characterizations are provided for blow-ups of a finite or an infinite Boolean graph (respectively, a preatomic graph). Then the characterizations are used to study co-maximal ideal graphs that are blow-ups of Boolean graphs (pre-atomic graphs, respectively).

Graph properties of co-maximal ideal graphs of commutative rings

In this paper, we determine the diameters of graphs [Formula: see text] and [Formula: see text] for a ring R with infinitely many maximal ideals. We also use graph blow-up to give a complete classification of rings R whose graphs [Formula: see text] are non-empty planar graphs.

Implements of Graph Blow-Up in Co-Maximal Ideal Graphs

In this article, graph blow-up is used to study the implements of the co-maximal graph and the co-maximal ideal graph 𝒞(R) of a commutative ring R. We show that for a ring R with finitely many maximal ideals, the graph 𝒞(R) is a blow-up of a Boolean graph, and we give more detailed structure of the graphs and 𝒞(R) as blow-ups of Boolean graphs. We determine for what rings R, 𝒞(R) or is a finite graph, respectively. In the end, we get some new results for rings by taking advantage of the graph .

CO-MAXIMAL IDEAL GRAPHS OF COMMUTATIVE RINGS

In this paper, a new kind of graph on a commutative ring R with identity, namely the co-maximal ideal graph is defined and studied. We use [Formula: see text] to denote this graph, with its vertices the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. We show some properties of this graph. For example, this graph is a simple, connected graph with diameter less than or equal to three, and both the clique number and the chromatic number of the graph are equal to the number of maximal ideals of the ring R.