Finite rings with complete bipartite zero-divisor graphs

2012 ◽  
Vol 56 (3) ◽  
pp. 20-25
Author(s):  
A. S. Kuzmina ◽  
Yu. N. Maltsev
Keyword(s):  
Zero Divisor ◽  
Finite Rings

FINITE RINGS WITH EULERIAN ZERO-DIVISOR GRAPHS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250055 ◽  
Author(s):  
A. S. KUZMINA
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Noncommutative Ring ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all non-zero (one-sided and two-sided) zero-divisors of R, and two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. [S. P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Commut. Rings1(4) (2002) 203–211.] In the present paper, all finite rings with Eulerian zero-divisor graphs are described.

Zero Divisor Graphs of Finite Direct Products of Finite Rings

2014 ◽  
Vol 42 (9) ◽  
pp. 3852-3860 ◽  
Author(s):  
Leah M. Birch ◽  
Jeremy J. Thibodeaux ◽  
Ralph P. Tucci
Keyword(s):  
Zero Divisor ◽  
Direct Products ◽  
Finite Rings

scholarly journals ON VARIETIES OF RINGS WHOSE FINITE RINGS ARE DETERMINED BY THEIR ZERO-DIVISOR GRAPHS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250019 ◽  
Author(s):  
A. S. Kuzmina ◽  
Yu. N. Maltsev
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge if and only if either xy = 0 or yx = 0. In the present paper, we study some properties of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.

scholarly journals Automorphisms of Zero Divisor Graphs of Cube Radical Zero Completely Primary Finite Rings

2020 ◽  
pp. 83-90
Author(s):  
Lao Hussein Mude ◽  
Owino Maurice Oduor ◽  
Ojiema Michael Onyango
Keyword(s):  
Algebraic Structure ◽  
Zero Divisor ◽  
Finite Rings ◽  
Zero Divisors

One of the most interesting areas of research that has attracted the attention of many scholars are theory of zero divisor graphs. Most recent research have focused on properties of zero divisor graphs with little attention given on the automorphsisms, despite the fact that automorphisms are useful in interpreting the symmetries of algebraic structure. Let R be a commutative unital finite rings and Z(R) be its set of zero divisors. In this study, the automorphisms zero divisor graphs of such rings in which the product of any three zero divisor is zero has been determined.

Finite Rings Whose Graphs Have Clique Number Less than Five

2021 ◽  
Vol 28 (03) ◽  
pp. 533-540
Author(s):  
Qiong Liu ◽  
Tongsuo Wu ◽  
Jin Guo
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Clique Number ◽  
Commutative Rings ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Rings

Let [Formula: see text] be a commutative ring and [Formula: see text] be its zero-divisor graph. We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one, two, or three. Furthermore, if [Formula: see text] (each [Formula: see text] is local for [Formula: see text]), we also give algebraic characterizations of the ring [Formula: see text] when the clique number of [Formula: see text] is four.

The compressed zero-divisor graphs of order 4

2021 ◽  
pp. 2250179
Author(s):  
A. S. Monastyreva
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Finite Rings ◽  
Associative Rings ◽  
Siberian Math

In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we find all graphs containing four vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring.

Finite rings with some restrictions on zero-divisor graphs

2014 ◽  
Vol 58 (12) ◽  
pp. 41-50
Author(s):  
A. S. Kuz’mina ◽  
Yu. N. Mal’tsev
Keyword(s):  
Zero Divisor ◽  
Finite Rings

NILPOTENT FINITE RINGS WITH PLANAR ZERO-DIVISOR GRAPHS

2008 ◽  
Vol 01 (04) ◽  
pp. 565-574 ◽  
Author(s):  
A. S. KUZ'MINA ◽  
Yu. N. MALTSEV
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph with all vertices non-zero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge iff xy = 0 or yx = 0 ([10]). In the present paper, we describe all nilpotent finite rings with planar zero-divisor graphs.

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