scholarly journals On Perfect Co-Annihilating-Ideal Graph of a Commutative Artinian Ring

2021 ◽  
Vol 45 (01) ◽  
pp. 63-73
Author(s):  
S. M. SAADAT MIRGHADIM ◽  
M. J. NIKMEHR ◽  
R. NIKANDISH
Keyword(s):  
Commutative Ring ◽  
Artinian Ring ◽  
Artinian Rings ◽  
Vertex Set

Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by AR, is a graph whose vertex set is the set of all non-zero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann(I) ∩ Ann(J) = (0). In this paper, we characterize all Artinian rings for which both of the graphs AR and AR (the complement of AR), are chordal. Moreover, all Artinian rings whose AR (and thus AR) is perfect are characterized.

scholarly journals RINGS WHOSE ANNIHILATING-IDEAL GRAPHS HAVE POSITIVE GENUS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250049 ◽  
Author(s):  
F. ALINIAEIFARD ◽  
M. BEHBOODI
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Artinian Ring ◽  
Commutative Rings ◽  
Gorenstein Ring ◽  
Isomorphism Classes ◽  
Artinian Rings ◽  
Vertex Set ◽  
Positive Genus

Let R be a commutative ring and 𝔸(R) be the set of ideals with nonzero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). We investigate commutative rings R whose annihilating-ideal graphs have positive genus γ(𝔸𝔾(R)). It is shown that if R is an Artinian ring such that γ(𝔸𝔾(R)) < ∞, then either R has only finitely many ideals or (R, 𝔪) is a Gorenstein ring with maximal ideal 𝔪 and v.dimR/𝔪𝔪/𝔪2= 2. Also, for any two integers g ≥ 0 and q > 0, there are only finitely many isomorphism classes of Artinian rings R satisfying the conditions: (i) γ(𝔸𝔾(R)) = g and (ii) |R/𝔪| ≤ q for every maximal ideal 𝔪 of R. Also, it is shown that if R is a non-domain Noetherian local ring such that γ(𝔸𝔾(R)) < ∞, then either R is a Gorenstein ring or R is an Artinian ring with only finitely many ideals.

scholarly journals On the Nonorientable Genus of the Generalized Unit and Unitary Cayley Graphs of a Commutative Ring

2022 ◽  
Vol 29 (01) ◽  
pp. 167-180
Author(s):  
Mahdi Reza Khorsandi ◽  
Seyed Reza Musawi
Keyword(s):  
Commutative Ring ◽  
Multiplicative Group ◽  
Cayley Graphs ◽  
Artinian Ring ◽  
Finite Rings ◽  
Artinian Rings ◽  
Vertex Set ◽  
Genus Formula ◽  
Multiplicative Subgroup ◽  
Unitary Cayley Graph

Let [Formula: see text] be a commutative ring and [Formula: see text] the multiplicative group of unit elements of [Formula: see text]. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, [Formula: see text], corresponding to a multiplicative subgroup [Formula: see text] of [Formula: see text] and a nonempty subset [Formula: see text] of [Formula: see text] with [Formula: see text], as the graph with vertex set [Formula: see text]and two distinct vertices [Formula: see text] and [Formula: see text] being adjacent if and only if there exists [Formula: see text] such that [Formula: see text]. In this paper, we characterize all Artinian rings [Formula: see text] for which [Formula: see text] is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring [Formula: see text] for which [Formula: see text] has finite nonorientable genus, [Formula: see text] must be a finite ring. Finally, it is proved that for a given positive integer [Formula: see text], the number of finite rings [Formula: see text] for which [Formula: see text] has nonorientable genus [Formula: see text] is finite.

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

2014 ◽  
Vol 21 (02) ◽  
pp. 249-256 ◽  
Author(s):  
G. Aalipour ◽  
S. Akbari ◽  
M. Behboodi ◽  
R. Nikandish ◽  
M. J. Nikmehr ◽  
...  
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Vertex Set

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. 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). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

Classification of rings with toroidal co-annihilating-ideal graphs

2020 ◽  
pp. 2150006
Author(s):  
Rana Khoeilar ◽  
Jafar Amjadi
Keyword(s):  
Commutative Ring ◽  
Artinian Rings ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity. The co-annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph 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]. In this paper, we study the planarity and genus of [Formula: see text]. In particular, we characterize all Artinian rings [Formula: see text] for which the genus of [Formula: see text] is zero or one.

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.

On the nil-graph of ideals of commutative Artinian rings

2018 ◽  
Vol 10 (05) ◽  
pp. 1850062
Author(s):  
T. Tamizh Chelvam ◽  
K. Selvakumar ◽  
P. Subbulakshmi
Keyword(s):  
Commutative Ring ◽  
Artinian Rings ◽  
Nilpotent Elements ◽  
Genus 2 ◽  
Vertex Set ◽  
The Ideal

Let [Formula: see text] be a commutative ring with identity and Nil[Formula: see text] be the ideal consisting of all nilpotent elements of [Formula: see text]. Let [Formula: see text] [Formula: see text] [Formula: see text]. The nil-graph of ideals of [Formula: see text] is defined as the graph [Formula: see text] whose vertex set is the set [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 discuss some properties of nil-graph of ideals concerning connectedness, split and claw free. Also we characterize all commutative Artinian rings [Formula: see text] for which the nil-graph [Formula: see text] has genus 2.

Cayley sum graph of ideals of commutative rings

2018 ◽  
Vol 17 (07) ◽  
pp. 1850125
Author(s):  
T. Tamizh Chelvam ◽  
K. Selvakumar ◽  
V. Ramanathan
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Artinian Rings ◽  
Vertex Set ◽  
Hamiltonian Properties ◽  
Cayley Sum Graph

Let [Formula: see text] be a commutative ring, [Formula: see text] the set of all ideals of [Formula: see text] and [Formula: see text], a subset of [Formula: see text]. The Cayley sum graph of ideals of [Formula: see text], denoted by Cay[Formula: see text], is a simple undirected graph with vertex set is the set [Formula: see text] and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text], for some [Formula: see text] in [Formula: see text]. In this paper, we study connectedness, Eulerian and Hamiltonian properties of Cay[Formula: see text]. Moreover, we characterize all commutative Artinian rings [Formula: see text] whose Cay[Formula: see text] is toroidal.

S-Artinian rings and finitely S-cogenerated rings

2019 ◽  
Vol 19 (03) ◽  
pp. 2050051 ◽  
Author(s):  
Esra Sengelen Sevim ◽  
Unsal Tekir ◽  
Suat Koc
Keyword(s):  
Commutative Ring ◽  
Finite Subset ◽  
Closed Subset ◽  
Artinian Ring ◽  
Index Set ◽  
Artinian Rings ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be a multiplicatively closed subset. In this paper, we study [Formula: see text]-Artinian rings and finitely [Formula: see text]-cogenerated rings. A commutative ring [Formula: see text] is said to be an [Formula: see text]-Artinian ring if for each descending chain of ideals [Formula: see text] of [Formula: see text] there exist [Formula: see text] and [Formula: see text] such that [Formula: see text] for all [Formula: see text] Also, [Formula: see text] is called a finitely [Formula: see text]-cogenerated ring if for each family of ideals [Formula: see text] of [Formula: see text] where [Formula: see text] is an index set, [Formula: see text] implies [Formula: see text] for some [Formula: see text] and a finite subset [Formula: see text] Moreover, we characterize some special rings such as Artinian rings and finitely cogenerated rings. Also, we extend many properties of Artinian rings and finitely cogenerated rings to [Formula: see text]-Artinian rings and finitely [Formula: see text]-cogenerated rings.

On projective intersection graph of ideals of commutative rings

2019 ◽  
pp. 2150017
Author(s):  
V. Ramanathan
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Principal Ideal ◽  
Artinian Ring ◽  
Intersection Graph ◽  
Principal Ideal Ring ◽  
Ideal Formula ◽  
Ideal Ring ◽  
Projective Graph ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] the set of all nontrivial proper ideals of [Formula: see text]. The intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as the set [Formula: see text], and, for any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some connections between commutative ring theory and graph theory by investigating topological properties of intersection graph of ideals. In particular, it is shown that for any nonlocal Artinian ring [Formula: see text], [Formula: see text] is a projective graph if and only if [Formula: see text] where [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] of nilpotency three and [Formula: see text] is a field. Furthermore, it is shown that for an Artinian ring [Formula: see text] [Formula: see text] if and only if [Formula: see text] where each [Formula: see text] is a local principal ideal ring with maximal ideal [Formula: see text] such that [Formula: see text]

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