scholarly journals On the extended zero divisor graph of commutative rings

2016 ◽  
Vol 40 ◽  
pp. 376-388 ◽  
Author(s):  
Driss BENNIS ◽  
Jilali MIKRAM ◽  
Fouad TARAZA
Keyword(s):  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph

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)).

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.

Characterizations of Three Classes of Zero-Divisor Graphs

2012 ◽  
Vol 55 (1) ◽  
pp. 127-137 ◽  
Author(s):  
John D. LaGrange
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Central Vertex ◽  
Integral Domains ◽  
Boolean Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

AbstractThe zero-divisor graph Γ(R) of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy = 0. In this paper, a characterization is provided for zero-divisor graphs of Boolean rings. Also, commutative rings R such that Γ(R) is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as those whose zero-divisor graphs are central vertex complete.

On Zero-divisor Graphs Whose Cores Contain no Rectangles

2011 ◽  
Vol 18 (04) ◽  
pp. 675-684 ◽  
Author(s):  
Qiong Liu ◽  
Tongsuo Wu
Keyword(s):  
Zero Divisor ◽  
Commutative Rings ◽  
Star Graph ◽  
Zero Divisor Graph ◽  
Isolated Vertex ◽  
Fan Graph

This paper answers a question of Lu and Wu: How can one characterize the zero-divisor graphs which contain no rectangles? We prove that a graph which contains no rectangles is the zero-divisor graph of a semigroup with 0 if and only if it is one of the following graphs: an isolated vertex, a star graph, a two-star graph, a triangle with n horns (n = 0, 1, 2, 3), a fan graph, a fan graph with a horn adjacent to its center. In addition, we completely determine the correspondence between commutative rings and finite zero-divisor graphs which contain no rectangles.

scholarly journals THE ZERO-DIVISOR GRAPHS OF RINGS AND SEMIRINGS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250033 ◽  
Author(s):  
DAVID DOLŽAN ◽  
POLONA OBLAK
Keyword(s):  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph

In this paper we study zero-divisor graphs of rings and semirings. We show that all zero-divisor graphs of (possibly noncommutative) semirings are connected and have diameter less than or equal to 3. We characterize all acyclic zero-divisor graphs of semirings and prove that in the case zero-divisor graphs are cyclic, their girths are less than or equal to 4. We find all possible cyclic zero-divisor graphs over commutative semirings having at most one 3-cycle, and characterize all complete k-partite and regular zero-divisor graphs. Moreover, we characterize all additively cancellative commutative semirings and all commutative rings such that their zero-divisor graph has exactly one 3-cycle.

scholarly journals On the Ideal Based Zero Divisor Graphs of Unital Commutative Rings and Galois Ring Module Idealizations

2021 ◽  
pp. 1-5
Author(s):  
Owino Maurice Oduor
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Galois Ring ◽  
Zero Divisor Graph ◽  
The Ideal

Let R be a commutative ring with identity 1 and I is an ideal of R. The zero divisor graph of the ring with respect to ideal has vertices defined as follows: {u ∈ Ic | uv ∈ I for some v ∈ Ic}, where Ic is the complement of I and two distinct vertices are adjacent if and only if their product lies in the ideal. In this note, we investigate the conditions under which the zero divisor graph of the ring with respect to the ideal coincides with the zero divisor graph of the ring modulo the ideal. We also consider a case of Galois ring module idealization and investigate its ideal based zero divisor graph.

scholarly journals Metric and upper dimension of zero divisor graphs associated to commutative rings

2020 ◽  
Vol 12 (1) ◽  
pp. 84-101 ◽  
Author(s):  
S. Pirzada ◽  
M. Aijaz
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Metric Dimension ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Finite Commutative Rings

AbstractLet R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}}, n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

Diameter and girth of the multiplicative zero-divisor graph of multiplicative lattices

2016 ◽  
Vol 09 (04) ◽  
pp. 1650071
Author(s):  
Vinayak Joshi ◽  
Sachin Sarode
Keyword(s):  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Multiplicative Lattice ◽  
Minimal Prime ◽  
Prime Elements ◽  
Multiplicative Lattices

In this paper, we study the multiplicative zero-divisor graph [Formula: see text] of a multiplicative lattice [Formula: see text]. Under certain conditions, we prove that for a reduced multiplicative lattice [Formula: see text] having more than two minimal prime elements, [Formula: see text] contains a cycle and [Formula: see text]. This essentially settles the conjecture of Behboodi and Rakeei [The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10(4) (2011) 741–753]. Further, we have characterized the diameter of [Formula: see text].

An ideal-based zero-divisor graph of direct products of commutative rings

2014 ◽  
Vol 34 (1) ◽  
pp. 45
Author(s):  
S. Ebrahimi Atani ◽  
M. Shajari Kohan ◽  
Z. Ebrahimi Sarvandi
Keyword(s):  
Zero Divisor ◽  
Commutative Rings ◽  
Direct Products ◽  
Zero Divisor Graph
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