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.

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.

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.

A new graph associated to a commutative ring

2016 ◽  
Vol 08 (02) ◽  
pp. 1650029 ◽  
Author(s):  
A. Alilou ◽  
J. Amjadi ◽  
S. M. Sheikholeslami
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Connected Graph ◽  
Commutative Rings ◽  
Simple Graph ◽  
Maximal Ideals ◽  
Vertex Set

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.

scholarly journals On the metric dimension of strongly annihilating-ideal graphs of commutative rings

2020 ◽  
Vol 12 (2) ◽  
pp. 358-369
Author(s):  
V. Soleymanivarniab ◽  
R. Nikandish ◽  
A. Tehranian
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Metric Dimension ◽  
Vertex Set

AbstractLet 𝒭 be a commutative ring with identity and 𝒜(𝒭) be the set of ideals with non-zero annihilator. The strongly annihilating-ideal graph of 𝒭 is defined as the graph SAG(𝒭) with the vertex set 𝒜 (𝒭)* = 𝒜 (𝒭) \{0} and two distinct vertices I and J are adjacent if and only if I ∩ Ann(J) ≠ (0) and J ∩ Ann(I) ≠ (0). In this paper, we study the metric dimension of SAG(𝒭) and some metric dimension formulae for strongly annihilating-ideal graphs are given.

Ideal-Based k-Zero-Divisor Hypergraph of Commutative Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 655-672
Author(s):  
K. Selvakumar ◽  
M. Subajini
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Zero Divisor ◽  
Necessary Conditions ◽  
Commutative Rings ◽  
Nonzero Ideal ◽  
Fixed Integer ◽  
Zero Divisors ◽  
Vertex Set ◽  
The Ideal

Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] a fixed integer. The ideal-based [Formula: see text]-zero-divisor hypergraph [Formula: see text] of [Formula: see text] has vertex set [Formula: see text], the set of all ideal-based [Formula: see text]-zero-divisors of [Formula: see text], and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge in [Formula: see text] if and only if [Formula: see text] and the product of the elements of any [Formula: see text]-subset of [Formula: see text] is not in [Formula: see text]. In this paper, we show that [Formula: see text] is connected with diameter at most 4 provided that [Formula: see text] for all ideal-based 3-zero-divisor hypergraphs. Moreover, we find the chromatic number of [Formula: see text] when [Formula: see text] is a product of finite fields. Finally, we find some necessary conditions for a finite ring [Formula: see text] and a nonzero ideal [Formula: see text] of [Formula: see text] to have [Formula: see text] planar.

scholarly journals A note on universally zero-divisor rings

1991 ◽  
Vol 43 (2) ◽  
pp. 233-239 ◽  
Author(s):  
S. Visweswaran
Keyword(s):  
Zero Divisor ◽  
Chain Condition ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Prüfer Domain ◽  
Integral Extension ◽  
Maximal Ideals ◽  
Unitary Module ◽  
Finite Set ◽  
Descending Chain Condition

In this note we consider commutative rings with identity over which every unitary module is a zero-divisor module. We call such rings Universally Zero-divisor (UZD) rings. We show (1) a Noetherian ring R is a UZD if and only if R is semilocal and the Krull dimension of R is at most one, (2) a Prüfer domain R is a UZD if and only if R has only a finite number of maximal ideals, and (3) if a ring R has Noetherian spectrum and descending chain condition on prime ideals then R is a UZD if and only if Spec (R) is a finite set. The question of ascent and descent of the property of a ring being a UZD with respect to integral extension of rings has also been answered.

The metric dimension of annihilator graphs of commutative rings

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.

scholarly journals Computing metric dimension of compressed zero divisor graphs associated to rings

2018 ◽  
Vol 10 (2) ◽  
pp. 298-318
Author(s):  
S. Pirzada ◽  
M. Imran Bhat
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Metric Dimension ◽  
Zero Divisor Graph ◽  
The Family ◽  
Vertex Set ◽  
Relationship Of ◽  
The Relationship

Abstract For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph ΓE(R) with vertex set Z(RE) \ {[0]} = RE \ {[0], [1]} defined by RE = {[x] : x ∈ R}, where [x] = {y ∈ R : ann(x) = ann(y)} and the two distinct vertices [x] and [y] of Z(RE) are adjacent if and only if [x][y] = [xy] = [0], that is, if and only if xy = 0. In this paper, we study the metric dimension of the compressed zero divisor graph ΓE(R), the relationship of metric dimension between ΓE(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of ΓE(R). We provide a formula for the number of vertices of the family of graphs given by ΓE(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of ΓE(R).

scholarly journals Domination number in the annihilating-ideal graphs of commutative rings

2015 ◽  
Vol 97 (111) ◽  
pp. 225-231 ◽  
Author(s):  
Reza Nikandish ◽  
Hamid Maimani ◽  
Sima Kiani
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Domination Number ◽  
Commutative Rings ◽  
Vertex Set ◽  
Domination Numbers

Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. 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. In this paper, we study the domination number of AG(R) and some connections between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are given.

N-ideals of commutative rings

Filomat ◽  
2017 ◽  
Vol 31 (10) ◽  
pp. 2933-2941 ◽  
Author(s):  
Unsal Tekir ◽  
Suat Koc ◽  
Kursat Oral
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Proper Ideal ◽  
Prime Ideals ◽  
Nonzero Identity

In this paper, we present a new classes of ideals: called n-ideal. Let R be a commutative ring with nonzero identity. We define a proper ideal I of R as an n-ideal if whenever ab ? I with a ? ?0, then b ? I for every a,b ? R. We investigate some properties of n-ideals analogous with prime ideals. Also, we give many examples with regard to n-ideals.

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