scholarly journals On compressed zero-divisor graphs of finite commutative local rings

2021 ◽  
Vol 18 (2) ◽  
pp. 1531-1555
Author(s):  
E. V. Zhuravlev ◽  
O. A. Filina
Keyword(s):  
Zero Divisor ◽  
Local Rings

scholarly journals Correction to: Classification of non-local rings with genus two zero-divisor graphs

2021 ◽  
Vol 25 (4) ◽  
pp. 3355-3356
Author(s):  
T. Asir ◽  
K. Mano ◽  
T. Tamizh Chelvam
Keyword(s):  
Zero Divisor ◽  
Local Rings ◽  
Non Local ◽  
Genus Two

Classification of non-local rings with genus two zero-divisor graphs

2019 ◽  
Vol 24 (1) ◽  
pp. 237-245
Author(s):  
T. Asir ◽  
K. Mano
Keyword(s):  
Zero Divisor ◽  
Local Rings ◽  
Non Local ◽  
Genus Two

Divisor graph of the complement of Γ(R)

2019 ◽  
Vol 12 (04) ◽  
pp. 1950057
Author(s):  
Ravindra Kumar ◽  
Om Prakash
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Integral Domain ◽  
Zero Divisor ◽  
Local Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring

Let [Formula: see text] be the complement of the zero-divisor graph of a finite commutative ring [Formula: see text]. In this paper, we provide the answer of the question (ii) raised by Osba and Alkam in [11] and prove that [Formula: see text] is a divisor graph if [Formula: see text] is a local ring. It is shown that when [Formula: see text] is a product of two local rings, then [Formula: see text] is a divisor graph if one of them is an integral domain. Further, if [Formula: see text], then [Formula: see text] is a divisor graph.

scholarly journals Categorial properties of compressed zero-divisor graphs of finite commutative rings

2020 ◽  
pp. 2150069
Author(s):  
Alen Đurić ◽  
Sara Jevđnić ◽  
Nik Stopar
Keyword(s):  
Principal Ideal ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Local Rings ◽  
Unital Ring ◽  
Zero Divisor Graph ◽  
Definition Of ◽  
Unital Rings ◽  
Finite Commutative Rings

By modifying the existing definition of a compressed zero-divisor graph [Formula: see text], we define a compressed zero-divisor graph [Formula: see text] of a finite commutative unital ring [Formula: see text], where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of [Formula: see text]). We prove that this is the best possible compression which induces a functor [Formula: see text], and that this functor preserves categorial products (in both directions). We use the structure of [Formula: see text] to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.

Classification of non-local rings with projective 3-zero-divisor hypergraph

2016 ◽  
Vol 66 (2) ◽  
pp. 457-468 ◽  
Author(s):  
K. Selvakumar ◽  
V. Ramanathan
Keyword(s):  
Zero Divisor ◽  
Local Rings ◽  
Non Local

scholarly journals On zero divisor graphs of finite commutative local rings

2019 ◽  
Vol 16 ◽  
pp. 465-480
Author(s):  
E. V. Zhuravlev ◽  
A. S. Monastyreva
Keyword(s):  
Zero Divisor ◽  
Local Rings

On Finite Local Rings with Clique Number Four

2022 ◽  
Vol 29 (01) ◽  
pp. 23-38
Author(s):  
Qiong Liu ◽  
Tongsuo Wu ◽  
Jin Guo
Keyword(s):  
Algebraic Structure ◽  
Zero Divisor ◽  
Clique Number ◽  
Local Rings ◽  
Zero Divisor Graph

We study the algebraic structure of rings [Formula: see text] whose zero-divisor graph [Formula: see text]has clique number four. Furthermore, we give complete characterizations of all the finite commutative local rings with clique number 4.

scholarly journals A class of zero divisor rings in which every graph is precisely the union of a complete graph and a complete bipartite graph

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Syed Khalid Nauman ◽  
Basmah H. Shafee
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Zero Divisor ◽  
Complete Bipartite Graph ◽  
Upper Bounds ◽  
Local Rings ◽  
Lower And Upper Bounds ◽  
Zero Divisor Graph ◽  
Characteristic Two

AbstractRecently, an interest is developed in estimating genus of the zero-divisor graph of a ring. In this note we investigate genera of graphs of a class of zero-divisor rings (a ring in which every element is a zero divisor). We call a ring R to be right absorbing if for a; b in R, ab is not 0, then ab D a. We first show that right absorbing rings are generalized right Klein 4-rings of characteristic two and that these are non-commutative zero-divisor local rings. The zero-divisor graph of such a ring is proved to be precisely the union of a complete graph and a complete bipartite graph. Finally, we have estimated lower and upper bounds of the genus of such a ring.

Local Rings with Genus Two Zero Divisor Graph

2010 ◽  
Vol 38 (8) ◽  
pp. 2965-2980 ◽  
Author(s):  
Nathan Bloomfield ◽  
Cameron Wickham
Keyword(s):  
Zero Divisor ◽  
Local Rings ◽  
Zero Divisor Graph ◽  
Genus Two
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