On Finite Local Rings with Clique Number Four

2022 ◽  
Vol 29 (01) ◽  
pp. 23-38
Author(s):  
Qiong Liu ◽  
Tongsuo Wu ◽  
Jin Guo
Keyword(s):  
Algebraic Structure ◽  
Zero Divisor ◽  
Clique Number ◽  
Local Rings ◽  
Zero Divisor Graph

We study the algebraic structure of rings [Formula: see text] whose zero-divisor graph [Formula: see text]has clique number four. Furthermore, we give complete characterizations of all the finite commutative local rings with clique number 4.

A generalized ideal-based zero-divisor graph

2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
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 ◽  
Vertex Set

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.

A ZERO-DIVISOR GRAPH FOR MODULES WITH RESPECT TO THEIR (FIRST) DUAL

2012 ◽  
Vol 12 (02) ◽  
pp. 1250151 ◽  
Author(s):  
M. BAZIAR ◽  
E. MOMTAHAN ◽  
S. SAFAEEYAN
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Complete Graph ◽  
Projective Module ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Covariant Functor ◽  
Zero Divisor Graph

Let M be an R-module. We associate an undirected graph Γ(M) to M in which nonzero elements x and y of M are adjacent provided that xf(y) = 0 or yg(x) = 0 for some nonzero R-homomorphisms f, g ∈ Hom (M, R). We observe that over a commutative ring R, Γ(M) is connected and diam (Γ(M)) ≤ 3. Moreover, if Γ(M) contains a cycle, then gr (Γ(M)) ≤ 4. Furthermore if ∣Γ(M)∣ ≥ 1, then Γ(M) is finite if and only if M is finite. Also if Γ(M) = ∅, then any nonzero f ∈ Hom (M, R) is monic (the converse is true if R is a domain). For a nonfinitely generated projective module P we observe that Γ(P) is a complete graph. We prove that for a domain R the chromatic number and the clique number of Γ(M) are equal. When R is self-injective, we will also observe that the above adjacency defines a covariant functor between a subcategory of R-MOD and the Category of graphs.

scholarly journals Thek-Zero-Divisor Hypergraph of a Commutative Ring

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Ch. Eslahchi ◽  
A. M. Rahimi
Keyword(s):  
Commutative Ring ◽  
Nilpotent Element ◽  
Zero Divisor ◽  
Clique Number ◽  
Common Point ◽  
Prime Ideals ◽  
Uniform Hypergraph ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and thek-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. LetRbe a commutative ring andkan integer strictly larger than2. Ak-uniform hypergraphHk(R)with the vertex setZ(R,k), the set of allk-zero-divisors inR, is associated toR, where eachk-subset ofZ(R,k)that satisfies thek-zero-divisor condition is an edge inHk(R). It is shown that ifRhas two prime idealsP1andP2with zero their only common point, thenHk(R)is a bipartite (2-colorable) hypergraph with partition setsP1−Z′andP2−Z′, whereZ′is the set of all zero divisors ofRwhich are notk-zero-divisors inR. IfRhas a nonzero nilpotent element, then a lower bound for the clique number ofH3(R)is found. Also, we have shown thatH3(R)is connected with diameter at most 4 wheneverx2≠0for all3-zero-divisorsxofR. Finally, it is shown that for any finite nonlocal ringR, the hypergraphH3(R)is complete if and only ifRis isomorphic toZ2×Z2×Z2.

ON THE COMPLEMENT OF THE ZERO-DIVISOR GRAPH OF A PARTIALLY ORDERED SET

2017 ◽  
Vol 97 (2) ◽  
pp. 185-193 ◽  
Author(s):  
SARIKA DEVHARE ◽  
VINAYAK JOSHI ◽  
JOHN LAGRANGE
Keyword(s):  
Partially Ordered Set ◽  
Zero Divisor ◽  
Ordered Set ◽  
Clique Number ◽  
Intersection Graph ◽  
Intersection Graphs ◽  
Partially Ordered ◽  
Zero Divisor Graph

In this paper, it is proved that the complement of the zero-divisor graph of a partially ordered set is weakly perfect if it has finite clique number, completely answering the question raised by Joshi and Khiste [‘Complement of the zero divisor graph of a lattice’,Bull. Aust. Math. Soc. 89(2014), 177–190]. As a consequence, the intersection graph of an intersection-closed family of nonempty subsets of a set is weakly perfect if it has finite clique number. These results are applied to annihilating-ideal graphs and intersection graphs of submodules.

Divisor graph of the complement of Γ(R)

2019 ◽  
Vol 12 (04) ◽  
pp. 1950057
Author(s):  
Ravindra Kumar ◽  
Om Prakash
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Integral Domain ◽  
Zero Divisor ◽  
Local Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring

Let [Formula: see text] be the complement of the zero-divisor graph of a finite commutative ring [Formula: see text]. In this paper, we provide the answer of the question (ii) raised by Osba and Alkam in [11] and prove that [Formula: see text] is a divisor graph if [Formula: see text] is a local ring. It is shown that when [Formula: see text] is a product of two local rings, then [Formula: see text] is a divisor graph if one of them is an integral domain. Further, if [Formula: see text], then [Formula: see text] is a divisor graph.

Finite Rings Whose Graphs Have Clique Number Less than Five

2021 ◽  
Vol 28 (03) ◽  
pp. 533-540
Author(s):  
Qiong Liu ◽  
Tongsuo Wu ◽  
Jin Guo
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Clique Number ◽  
Commutative Rings ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Rings

Let [Formula: see text] be a commutative ring and [Formula: see text] be its zero-divisor graph. We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one, two, or three. Furthermore, if [Formula: see text] (each [Formula: see text] is local for [Formula: see text]), we also give algebraic characterizations of the ring [Formula: see text] when the clique number of [Formula: see text] is four.

ZERO-DIVISOR GRAPH OF AN IDEAL OF A NEAR-RING

2013 ◽  
Vol 05 (01) ◽  
pp. 1350007 ◽  
Author(s):  
T. TAMIZH CHELVAM ◽  
S. NITHYA
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Clique Number ◽  
Zero Divisor Graph

In this paper, we associate the graph ΓI(N) to an ideal I of a near-ring N. We exhibit some properties and structure of ΓI(N). For a commutative ring R, Beck conjectured that both chromatic number and clique number of the zero-divisor graph Γ(R) of R are equal. We prove that Beck's conjecture is true for ΓI(N). Moreover, we characterize all right permutable near-rings N for which the graph ΓI(N) is finitely colorable.

On the essential graph of a commutative ring

2016 ◽  
Vol 16 (07) ◽  
pp. 1750132 ◽  
Author(s):  
M. J. Nikmehr ◽  
R. Nikandish ◽  
M. Bakhtyiari
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Artinian Ring ◽  
Clique Number ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Essential Ideal ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity, and let [Formula: see text] be the set of zero-divisors of [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 ann[Formula: see text] is an essential ideal. It is proved that [Formula: see text] is connected with diameter at most three and with girth at most four, if [Formula: see text] contains a cycle. Furthermore, rings with complete or star essential graphs are characterized. Also, we study the affinity between essential graph and zero-divisor graph that is associated with a ring. Finally, we show that the essential graph associated with an Artinian ring is weakly perfect, i.e. its vertex chromatic number equals its clique number.

scholarly journals Categorial properties of compressed zero-divisor graphs of finite commutative rings

2020 ◽  
pp. 2150069
Author(s):  
Alen Đurić ◽  
Sara Jevđnić ◽  
Nik Stopar
Keyword(s):  
Principal Ideal ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Local Rings ◽  
Unital Ring ◽  
Zero Divisor Graph ◽  
Definition Of ◽  
Unital Rings ◽  
Finite Commutative Rings

By modifying the existing definition of a compressed zero-divisor graph [Formula: see text], we define a compressed zero-divisor graph [Formula: see text] of a finite commutative unital ring [Formula: see text], where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of [Formula: see text]). We prove that this is the best possible compression which induces a functor [Formula: see text], and that this functor preserves categorial products (in both directions). We use the structure of [Formula: see text] to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.

scholarly journals On the Line Graph for Zero-Divisors of C(X)

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Ghada AlAfifi ◽  
Emad Abu Osba
Keyword(s):  
Hausdorff Space ◽  
Zero Divisor ◽  
Line Graph ◽  
Clique Number ◽  
Completely Regular ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Dominating Number ◽  
Isolated Points

Let be X a completely regular Hausdorff space and let C(X) be the ring of all continuous real valued functions defined on X. In this paper, the line graph for the zero-divisor graph of C(X) is studied. It is shown that this graph is connected with diameter less than or equal to 3 and girth 3. It is shown that this graph is always triangulated and hypertriangulated. It is characterized when the graph is complemented. It is proved that the radius of this graph is 2 if and only if X has isolated points; otherwise, the radius is 3. Bounds for the dominating number and clique number are also found in terms of the density number of X.

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