scholarly journals A generalization of the zero-divisor graph for modules

2019 ◽  
Vol 106 (120) ◽  
pp. 39-46
Author(s):  
Katayoun Nozari ◽  
Shiroyeh Payrovi
Keyword(s):  
Commutative Ring ◽  
Complete Graph ◽  
Zero Divisor ◽  
Simple Graph ◽  
Zero Divisor Graph ◽  
Maximal Ideals

Let R be a commutative ring and M a Noetherian R-module. The zero-divisor graph of M, denoted by ?(M), is an undirected simple graph whose vertices are the elements of ZR(M)\AnnR(M) and two distinct vertices a and b are adjacent if and only if abM = 0. In this paper, we study diameter and girth of ?(M). We show that the zero-divisor graph of M has a universal vertex in ZR(M)\r(AnnR(M)) if and only if R = ?Z2?R? and M = Z2?M?, where M? is an R?-module. Moreover, we show that if ?(M) is a complete graph, then one of the following statements is true: (i) AssR(M) = {m1,m2}, where m1,m2 are maximal ideals of R. (ii) AssR(M) = {p}, where p 2 ? AnnR(M). (iii) AssR(M) = {p}, where p 3 ? AnnR(M).

ON RINGS R WHOSE GRAPHS Γ(R) SATISFY CONDITION (Kp)

2011 ◽  
Vol 10 (04) ◽  
pp. 665-674
Author(s):  
LI CHEN ◽  
TONGSUO WU
Keyword(s):  
Prime Number ◽  
Commutative Ring ◽  
Complete Graph ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Satisfy Condition ◽  
Induced Subgraph ◽  
Zero Divisor Graph ◽  
Ring Graph ◽  
Finite Commutative Rings

Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let CG(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph Kp-1, a vertex c ∈ CG(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (Kp)).

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 On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 482
Author(s):  
Bilal A. Rather ◽  
Shariefuddin Pirzada ◽  
Tariq A. Naikoo ◽  
Yilun Shang
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Simple Graph ◽  
Laplacian Spectrum ◽  
Laplacian Eigenvalues ◽  
Ring Of Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Positive Integers

Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖{0} be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p<q are primes and N1,N2 are positive integers.

scholarly journals On the Genus of the Zero-Divisor Graph of Zn

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Huadong Su ◽  
Pailing Li
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Orientable Surface ◽  
Simple Graph ◽  
Ring Of Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors

Let R be a commutative ring with identity. The zero-divisor graph of R, denoted Γ(R), is the simple graph whose vertices are the nonzero zero-divisors of R, and two distinct vertices x and y are linked by an edge if and only if xy=0. The genus of a simple graph G is the smallest integer g such that G can be embedded into an orientable surface Sg. In this paper, we determine that the genus of the zero-divisor graph of Zn, the ring of integers modulo n, is two or three.

scholarly journals On the diameter of the graph ГAnn(M)(R)

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 623-629 ◽  
Author(s):  
David Anderson ◽  
Shaban Ghalandarzadeh ◽  
Sara Shirinkam ◽  
Parastoo Rad
Keyword(s):  
Commutative Ring ◽  
Complete Graph ◽  
Zero Divisor ◽  
Prime Ideals ◽  
Annihilator Ideal ◽  
Zero Divisor Graph ◽  
Minimal Prime ◽  
The Relationship ◽  
The Ideal

For a commutative ring R with identity, the ideal-based zero-divisor graph, denoted by ?I (R), is the graph whose vertices are {x ? R\I|xy ? I for some y ? R\I}, and two distinct vertices x and y are adjacent if and only if xy?I. In this paper, we investigate an annihilator ideal-based zero-divisor graph, denoted by ?Ann(M)(R), by replacing the ideal I with the annihilator ideal Ann(M) for an R-module M. We also study the relationship between the diameter of ?Ann(M) (R) and the minimal prime ideals of Ann(M). In addition, we determine when ?Ann(M)(R) is complete. In particular, we prove that for a reduced R-module M, ?Ann(M) (R) is a complete graph if and only if R ? Z2?Z2 and M ? M1?M2 for M1 and M2 nonzero Z2-modules.

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.

Some properties of zero divisor graph obtained by the ring Zp × Zq × Zr

2019 ◽  
Vol 12 (06) ◽  
pp. 2040001
Author(s):  
Nihat Akgunes ◽  
Yasar Nacaroglu
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Topological Indices ◽  
Zero Divisor Graph

The concept of zero-divisor graph of a commutative ring was introduced by Beck [Coloring of commutating ring, J. Algebra 116 (1988) 208–226]. In this paper, we present some properties of zero divisor graphs obtained from ring [Formula: see text], where [Formula: see text] and [Formula: see text] are primes. Also, we give some degree-based topological indices of this special graph.

On zero-divisor graphs of commutative rings without identity

2019 ◽  
Vol 19 (12) ◽  
pp. 2050226 ◽  
Author(s):  
G. Kalaimurugan ◽  
P. Vignesh ◽  
T. Tamizh Chelvam
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring ◽  
Free Order ◽  
Finite Commutative Rings

Let [Formula: see text] be a finite commutative ring without identity. In this paper, we characterize all finite commutative rings without identity, whose zero-divisor graphs are unicyclic, claw-free and tree. Also, we obtain all finite commutative rings without identity and of cube-free order for which the corresponding zero-divisor graph is toroidal.

Annihilating-ideal graphs of commutative rings

2019 ◽  
Vol 13 (07) ◽  
pp. 2050121
Author(s):  
M. Aijaz ◽  
S. Pirzada
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Metric Dimension ◽  
Maximal Ideals ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with unity [Formula: see text]. The annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is defined to be the graph with vertex set [Formula: see text] — the set of non-zero annihilating ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] adjacent if and only if [Formula: see text]. Some connections between annihilating-ideal graphs and zero divisor graphs are given. We characterize the prime ideals (or equivalently maximal ideals) of [Formula: see text] in terms of their degrees as vertices of [Formula: see text]. We also obtain the metric dimension of annihilating-ideal graphs of commutative rings.

Associate class graph of zero-divisors of a commutative ring

2019 ◽  
Vol 19 (08) ◽  
pp. 2050155
Author(s):  
Gaohua Tang ◽  
Guangke Lin ◽  
Yansheng Wu
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
New Class ◽  
Associate Class ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Class Graph

In this paper, we introduce the concept of the associate class graph of zero-divisors of a commutative ring [Formula: see text], denoted by [Formula: see text]. Some properties of [Formula: see text], including the diameter, the connectivity and the girth are investigated. Utilizing this graph, we present a new class of counterexamples of Beck’s conjecture on the chromatic number of the zero-divisor graph of a commutative ring.

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