Metric dimension of complement of annihilator graphs associated with commutative rings

2021 ◽  
Author(s):  
Sh. Ebrahimi ◽  
R. Nikandish ◽  
A. Tehranian ◽  
H. Rasouli
Keyword(s):  
Commutative Rings ◽  
Metric Dimension

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 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.

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 On Certain Bounds for Edge Metric Dimension of Zero-Divisor Graphs Associated with Rings

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Hafiz Muahmmad Afzal Siddiqui ◽  
Ammar Mujahid ◽  
Muhammad Ahsan Binyamin ◽  
Muhammad Faisal Nadeem
Keyword(s):  
Zero Divisor ◽  
Research Work ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Metric Dimension ◽  
Gaussian Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Algebraic Properties ◽  
Do So

Given a finite commutative unital ring S having some non-zero elements x ,   y such that x . y = 0 , the elements of S that possess such property are called the zero divisors, denoted by Z S . We can associate a graph to S with the help of zero-divisor set Z S , denoted by ζ S (called the zero-divisor graph), to study the algebraic properties of the ring S . In this research work, we aim to produce some general bounds for the edge version of metric dimension regarding zero-divisor graphs of S . To do so, we will discuss the zero-divisor graphs for the ring of integers ℤ m modulo m , some quotient polynomial rings, and the ring of Gaussian integers ℤ m i modulo m . Then, we prove the general result for the bounds of edge metric dimension of zero-divisor graphs in terms of maximum degree and diameter of ζ S . In the end, we provide the commutative rings with the same metric dimension, edge metric dimension, and upper dimension.

A correspondence between commutative rings and Jordan loops

2020 ◽  
Vol 58 (6) ◽  
pp. 741-768
Author(s):  
V. I. Ursu
Keyword(s):  
Commutative Rings

GENERALIZED OF PRIMARY AVOIDANCE THEOREM FOR COMMUTATIVE RINGS

2018 ◽  
Vol 1 (21) ◽  
pp. 415-438
Author(s):  
Amer Shamil Abdulrhman
Keyword(s):  
Commutative Rings ◽  
Primary Ideals

In this paper we study covering ideals by Cosets of primary ideals and we get a generalized the primary avoidance theorem in the rings which it has been

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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