Maximal Ideal Graph of Commutative Rings

In this paper, we introduce and study the notion of the maximal ideal graph of a commutative ring with identity. Let R be a commutative ring with identity. The maximal ideal graph of R, denoted by MG(R), is the undirected graph with vertex set, the set of non-trivial ideals of R, where two vertices I1 and I2 are adjacent if I1 I2 and I1+I2 is a maximal ideal of R. We explore some of the properties and characterizations of the graph.

Crosscap two of class of graphs from commutative rings

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830922500410 ◽

2021 ◽

Author(s):

S. Karthik ◽

S. N. Meera ◽

K. Selvakumar

Keyword(s):

Commutative Ring ◽

Undirected Graph ◽

Commutative Rings ◽

Zero Divisors ◽

Essential Ideal ◽

Vertex Set ◽

Annihilator Graph ◽

Finite Commutative Rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The annihilator graph of commutative ring [Formula: see text] is the simple undirected graph [Formula: see text] with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal. In this paper, we classify all finite commutative rings with identity whose annihilator graph and essential graph have crosscap two.

The metric dimension of annihilator graphs of commutative rings

10.1142/s0219498820500899 ◽

2019 ◽

Vol 19 (05) ◽

pp. 2050089

Author(s):

V. Soleymanivarniab ◽

A. Tehranian ◽

R. Nikandish

Keyword(s):

Commutative Ring ◽

Undirected Graph ◽

Commutative Rings ◽

Metric Dimension ◽

Zero Divisors ◽

Vertex Set ◽

Annihilator Graph ◽

Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity. The annihilator graph of [Formula: see text], denoted by [Formula: see text], is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study the metric dimension of annihilator graphs associated with commutative rings and some metric dimension formulae for annihilator graphs are given.

THE ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS I

10.1142/s0219498811004896 ◽

2011 ◽

Vol 10 (04) ◽

pp. 727-739 ◽

Cited By ~ 69

Author(s):

M. BEHBOODI ◽

Z. RAKEEI

Keyword(s):

Commutative Ring ◽

Maximal Ideal ◽

Connected Graph ◽

Undirected Graph ◽

Chain Condition ◽

Commutative Rings ◽

Finiteness Conditions ◽

Descending Chain Condition ◽

Ascending Chain Condition

Let R be a commutative ring, with 𝔸(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). It is the (undirected) graph with vertices 𝔸(R)* ≔ 𝔸(R)\{(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of 𝔸𝔾(R). For instance, it is shown that if R is not a domain, then 𝔸𝔾(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of 𝔸𝔾(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, 𝔸𝔾(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of 𝔸𝔾(R). It is shown that 𝔸𝔾(R) is a connected graph and diam (𝔸𝔾)(R) ≤ 3 and if 𝔸𝔾(R) contains a cycle, then gr (𝔸𝔾(R)) ≤ 4. Also, rings R for which the graph 𝔸𝔾(R) is complete or star, are characterized, as well as rings R for which every vertex of 𝔸𝔾(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

RINGS WHOSE ANNIHILATING-IDEAL GRAPHS HAVE POSITIVE GENUS

10.1142/s0219498811005774 ◽

2012 ◽

Vol 11 (03) ◽

pp. 1250049 ◽

Cited By ~ 3

Author(s):

F. ALINIAEIFARD ◽

M. BEHBOODI

Keyword(s):

Commutative Ring ◽

Local Ring ◽

Maximal Ideal ◽

Artinian Ring ◽

Commutative Rings ◽

Gorenstein Ring ◽

Isomorphism Classes ◽

Artinian Rings ◽

Vertex Set ◽

Positive Genus

Let R be a commutative ring and 𝔸(R) be the set of ideals with nonzero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). We investigate commutative rings R whose annihilating-ideal graphs have positive genus γ(𝔸𝔾(R)). It is shown that if R is an Artinian ring such that γ(𝔸𝔾(R)) < ∞, then either R has only finitely many ideals or (R, 𝔪) is a Gorenstein ring with maximal ideal 𝔪 and v.dimR/𝔪𝔪/𝔪2= 2. Also, for any two integers g ≥ 0 and q > 0, there are only finitely many isomorphism classes of Artinian rings R satisfying the conditions: (i) γ(𝔸𝔾(R)) = g and (ii) |R/𝔪| ≤ q for every maximal ideal 𝔪 of R. Also, it is shown that if R is a non-domain Noetherian local ring such that γ(𝔸𝔾(R)) < ∞, then either R is a Gorenstein ring or R is an Artinian ring with only finitely many ideals.

Cayley sum graph of ideals of commutative rings

10.1142/s0219498818501256 ◽

2018 ◽

Vol 17 (07) ◽

pp. 1850125

Author(s):

T. Tamizh Chelvam ◽

K. Selvakumar ◽

V. Ramanathan

Keyword(s):

Commutative Ring ◽

Undirected Graph ◽

Commutative Rings ◽

Artinian Rings ◽

Vertex Set ◽

Hamiltonian Properties ◽

Cayley Sum Graph

Let [Formula: see text] be a commutative ring, [Formula: see text] the set of all ideals of [Formula: see text] and [Formula: see text], a subset of [Formula: see text]. The Cayley sum graph of ideals of [Formula: see text], denoted by Cay[Formula: see text], is a simple undirected graph with vertex set is the set [Formula: see text] and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text], for some [Formula: see text] in [Formula: see text]. In this paper, we study connectedness, Eulerian and Hamiltonian properties of Cay[Formula: see text]. Moreover, we characterize all commutative Artinian rings [Formula: see text] whose Cay[Formula: see text] is toroidal.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

10.1142/s021949881250199x ◽

2013 ◽

Vol 12 (04) ◽

pp. 1250199 ◽

Cited By ~ 11

Author(s):

T. ASIR ◽

T. TAMIZH CHELVAM

Keyword(s):

Commutative Ring ◽

Independence Number ◽

Clique Number ◽

Commutative Rings ◽

Intersection Graph ◽

Vertex Connectivity ◽

Total Graph ◽

Graph Theoretic ◽

Vertex Set ◽

Finite Commutative Rings

The intersection graph ITΓ(R) of gamma sets in the total graph TΓ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about ITΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on ITΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of ITΓ(R). Further, we focus on certain graph theoretic parameters of ITΓ(R) like the independence number, the clique number and the connectivity of ITΓ(R). Also, we obtain both vertex and edge chromatic numbers of ITΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(ITΓ(R)) = ω(ITΓ(R)). Having proved that ITΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which ITΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which ITΓ(R) is of class one (i.e. χ′(ITΓ(R)) = Δ(ITΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in ITΓ(R).

The Weakly Nilpotent Graph of a Commutative Ring

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-096-1 ◽

2017 ◽

Vol 60 (2) ◽

pp. 319-328

Author(s):

Soheila Khojasteh ◽

Mohammad Javad Nikmehr

Keyword(s):

Commutative Ring ◽

Chromatic Number ◽

Disjoint Union ◽

Independence Number ◽

Artinian Ring ◽

Clique Number ◽

Commutative Rings ◽

Unicyclic Graph ◽

Nilpotent Elements ◽

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

THE JACOBSON GRAPH OF COMMUTATIVE RINGS

10.1142/s0219498812501794 ◽

2012 ◽

Vol 12 (03) ◽

pp. 1250179 ◽

Cited By ~ 10

Author(s):

A. AZIMI ◽

A. ERFANIAN ◽

M. FARROKHI D. G.

Keyword(s):

Commutative Ring ◽

Chromatic Number ◽

Independence Number ◽

Commutative Rings ◽

Dominating Number ◽

Jacobson Graph ◽

Vertex Set ◽

Numerical Invariants ◽

Nonzero Identity

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 𝔍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number.

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

Algebra Colloquium ◽

10.1142/s1005386714000200 ◽

2014 ◽

Vol 21 (02) ◽

pp. 249-256 ◽

Cited By ~ 20

Author(s):

G. Aalipour ◽

S. Akbari ◽

M. Behboodi ◽

R. Nikandish ◽

M. J. Nikmehr ◽

...

Keyword(s):

Commutative Ring ◽

Chromatic Number ◽

Artinian Ring ◽

Clique Number ◽

Commutative Rings ◽

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

A generalized ideal-based zero-divisor graph

10.1142/s0219498815500796 ◽

2015 ◽

Vol 14 (06) ◽

pp. 1550079 ◽

Cited By ~ 1

Author(s):

M. J. Nikmehr ◽

S. Khojasteh

Keyword(s):

Prime Ideal ◽

Commutative Ring ◽

Chromatic Number ◽

Zero Divisor ◽

Undirected Graph ◽

Clique Number ◽

Proper Ideal ◽

Graph Structure ◽

Zero Divisor Graph ◽

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.