Planar of special idealization rings

Journal of Pure and Applied Algebra, 2006

We consider zero-divisor graphs of idealizations of commutative rings. Specifically, we look at the preservation, or lack thereof, of the diameter and girth of the zero-divisor graph of a ring when extending to idealizations of the ring.

2018

The rings considered in this article are nonzero commutative with identity which are not fields. Let \(R\) be a ring. We denote the collection of all proper ideals of \(R\) by \(\mathbb{I}(R)\) and the collection \(\mathbb{I}(R)\setminus \{(0)\}\) by \(\mathbb{I}(R)^{*}\). Recall that the intersection graph of ideals of \(R\), denoted by \(G(R)\), is an undirected graph whose vertex set is \(\mathbb{I}(R)^{*}\) and distinct vertices \(I, J\) are adjacent if and only if \(I\cap J\neq (0)\). In this article, we consider a subgraph of \(G(R)\), denoted by \(H(R)\), whose vertex set is \(\mathbb{I}(R)^{*}\) and distinct vertices \(I, J\) are adjacent in \(H(R)\) if and only if \(IJ\neq (0)\). The purpose of this article is to characterize rings \(R\) with at least two maximal ideals such that \(H(R)\) is planar.

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 C (R) is an undirected simple graph whose vertex set is the set of all proper ideals I of R such that I ̸⊆ J(R), where J(R) is the Jacobson radical of R and distinct vertices I1, I2 are joined by an edge in C(R) if and only if I1 + I2 = R. In Section 2 of this article, we classify rings R such that C (R) is planar. In Section 3 of this article, we classify rings R such that C (R) is a split graph. In Section 4 of this article, we classify rings R such that C(R) is complemented and moreover, we determine the S-vertices of C (R).

Bollettino dell'Unione Matematica Italiana, 2016

Let R be a commutative ring with identity. Let A(R) denote the collection of all annihilating ideals of R (that is, A(R) is the collection of all ideals I of R which admits a nonzero annihilator in R). Let AG(R) denote the annihilating ideal graph of R. In this article, necessary and sufficient conditions are determined in order that AG(R) is complemented under the assumption that R is a zero-dimensional quasisemilocal ring which admits at least two nonzero annihilating ideals and as a corollary we determine finite rings R such that AG(R) is complemented under the assumption that A(R) contains at least two nonzero ideals.

2012

In this paper, we determine the diameters of graphs Γ ′ 2 (R) and C (R) 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 C (R) are non-empty planar graphs.

Algebra Colloquium, 2014

Let R be a commutative ring and Max (R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by − − → Γreg(R), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γreg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian

2016

Let be a commutative ring with nonzero identity and be a proper ideal of . The annihilator graph of with respect to , which is denoted by , is the undirected graph with vertex-set for some and two distinct vertices and are adjacent if and only if , where . In this paper, we study some basic properties of , and we characterise when is planar, outerplanar or a ring graph. Also, we study the graph , where is the ring of integers modulo .

Ricerche di Matematica, 2016

Let R be a commutative ring with nonzero identity. We denote by AG(R) the annihilator graph of R, whose vertex set consists of the set of nonzero zero-divisors of R, and two distint vertices x and y are adjacent if and only if ann(x) ∪ ann(y) = ann(x y), where for t ∈ R, we set ann(t) := {r ∈ R | rt = 0}. In this paper, we define the annihiator-ideal graph of R, which is denoted by A I (R), as an undirected graph with vertex set A * (R), and two distinct vertices I and J are adjacent if and only if ann(I) ∪ ann(J) = ann(I J). We study some basic properties of A I (R) such as connectivity, diameter and girth. Also we investigate the situations under which the graphs AG(R) and A I (R) are coincide. Moreover, we examin the planarity of the graph A I (R).

Let $S$ be a semigroup with $0$ and $R$ be a ring with $1$. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}(S)$, and the other definition yields an undirected graph $\overline{{\Gamma}}(S)$. It is shown that $\Gamma(S)$ is not necessarily connected, but $\overline{{\Gamma}}(S)$ is always connected and ${\rm diam}(\overline{\Gamma}(S))\leq 3$. For a ring $R$ define a directed graph $\Bbb{APOG}(R)$ to be equal to $\Gamma(\Bbb{IPO}(R))$, where $\Bbb{IPO}(R)$ is a semigroup consisting of all products of two one-sided ideals of $R$, and define an undirected graph $\overline{\Bbb{APOG}}(R)$ to be equal to $\overline{\Gamma}(\Bbb{IPO}(R))$. We show that $R$ is an Artinian (resp., Noetherian) r...

Journal of Algebra, 2003

Let Γ(R) be the zero-divisor graph of a commutative ring R. An interesting question was proposed by Anderson, Frazier, Lauve, and Livingston: For which finite commutative rings R is Γ(R) planar? We give an answer to this question. More precisely, we prove that if R is a local ring with at least 33 elements, and Γ(R)≠∅, then Γ(R) is not planar.

Journal of Algebra and its Applications, 12(4)(2013), 1250198 (18 pages) (DOI: 10.1142/ S0219498812501988), 2013

Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra 320 (2008) 2706-2719] introduced the total graph of R, denoted by T Γ (R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a commutative ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of T Γ (R) where R is a commutative Artin ring. The intersection graph of gamma sets in T Γ (R) is denoted by I T Γ (R). Tamizh Chelvam and Asir [Intersection graph of gamma sets in the total graph, Discuss. Math. Graph Theory 32 (2012) 339-354, doi:10.7151/dmgt.1611] initiated a study about the intersection graph I T Γ (Zn) of gamma sets in T Γ (Zn). In this paper, we study about I T Γ (R), where R is a commutative Artin ring. Actually we investigate the interplay between graph-theoretic properties of I T Γ (R) and ring-theoretic properties of R. At the first instance, we prove that diam(I T Γ (R)) ≤ 2 and gr(I T Γ (R)) ≤ 4. Also some characterization results regarding completeness, bipartite, cycle and chordal nature of I T Γ (R) are given. Further, we discuss about the vertex-transitive property of I T Γ (R). At last, we obtain all commutative Artin rings R for which I T Γ (R) is either planar or toroidal or genus two.

2021

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Let R be a ring with unity and I(R)∗ are all non-trivial left ideals of R. The intersection graph of ideals of R is denoted by G(R) is an undirected simple graph with vertex set I(R)∗ and two distinct vertices I and J are adjacent if and only if I ∩ J 6= 0. In this article, we investigate some basic properties of the line graph associated to G(R), denoted by L(G(R)). Moreover, we investigate completeness, unicyclicness, bipartiteness, planarity, outerplanarity, ring graph, diameter, girth and clique of L(G(Zn)). We also investigate some basic properties of L(G(R)) for left Artinian ring and finally, we determine the domination number and bondage number of L(G(Zn)).

International Electronic Journal of Algebra

Let $R$ be a commutative ring without identity. The zero-divisor graph of $R,$ denoted by $\Gamma(R)$ is a graph with vertex set $Z(R)\setminus \{0\}$ which is the set of all nonzero zero-divisor elements of $R,$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0.$ In this paper, we characterize the rings whose zero-divisor graphs are ring graphs and outerplanar graphs. Further, we establish the planar index, ring index and outerplanar index of the zero-divisor graphs of finite commutative rings without identity.

Discrete Mathematics, Algorithms and Applications, 2016

Let [Formula: see text] be a commutative ring with identity. In this paper, we consider a simple graph associated with [Formula: see text] denoted by [Formula: see text], whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] or [Formula: see text]. In this paper, we initiate the study of the graph [Formula: see text] and we investigate its properties. In particular, we show that [Formula: see text] is a connected graph with [Formula: see text] unless [Formula: see text] is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings [Formula: see text] with at least two maximal ideals for which [Formula: see text] are planar.

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 I1 and I2 are adjacent if and only if I1 + I2 = 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. *