Article
Kyungpook Mathematical Journal 2020; 60(4): 723-729
Published online December 31, 2020
Copyright © Kyungpook Mathematical Journal.
The Zero-divisor Graph of
Min Ji Park, Eun Sup Kim, Jung Wook Lim*
Department of Mathematics, College of Life Science and Nano Technology, Hannam University, Daejeon 34430, Republic of Korea
e-mail : mjpark5764@gmail.com
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail : eskim@knu.ac.kr
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail : jwlim@knu.ac.kr
Received: June 13, 2020; Revised: July 28, 2020; Accepted: August 4, 2020
Abstract
Let
Keywords: Γ(ℤ,n[X)⟧,, diameter, girth, clique, chromatic number
1. Introduction
1.1. Preliminaries
In this subsection, we review some concepts from basic graph theory. Let
1.2. The Zero-divisor Graph of a Commutative Ring
Let
In [6], Beck first introduced the concept of the zero-divisor graph of a commutative ring and in [3], Anderson and Naseer continued to study. In [3] and [6], all elements of
For more on the zero-divisor graph of a commutative ring, the readers can refer to a survey article [1].
Let
2. The Diameter and the Girth of
In order to give the complete characterization of the diameter of
Lemma 2.1. ([4, Chapter 1, Exercise 2(iii)] and [7, Theorem 5])
-
(1) If
, then there exists a nonzero element r ∈ R such thatrf=0 . -
(2) If
R is a Noetherian ring and, then there exists a nonzero element r ∈ R such thatrf=0 .
Let
Theorem 2.2.
The following statements hold.
-
(1)
if ( and only if )n=p2 for some primep . -
(2)
if ( and only if )n=pr for some primep and some integerr ≥ 3 , orn=pq for some distinct primesp andq . -
(3)
if ( and only if )n=pqr for some distinct primesp, q and some integerr ≥ 2 .
(1) Suppose that
(2) Suppose that
Suppose that
(3) Suppose that
-
Figure 1. The diameter of some zero-divisor graphs
We next study the girth of
Proposition 2.3.
If
Proposition 2.4.
If
Lemma 2.5.
If
Lemma 2.6.
Let
Proposition 2.7.
Let
By Propositions 2.3, 2.4 and 2.7, we can completely characterize the girth of
Theorem 2.8.
The following statements hold.
-
(1)
if ( and only if ) each of the following conditions holds.-
(a)
n=pr for some primep and integerr ≥ 2 . -
(b)
n=pqr for some distinct primesp, q and integerr ≥ 2 .
-
-
(2)
if ( and only if )n=pq for some distinct primesp andq .
-
Figure 2. The girth of some zero-divisor graphs
3. The Chromatic Number of
In this section, we calculate the chromatic number of
Lemma 3.1.
If
Proposition 3.2.
If
To complete the proof, it remains to check that any two vertices of
Thus we conclude that
We denote the set of nonnegative integers by
Lemma 3.3.
If
By Cases 1 and 2,
Proposition 3.4.
Let
Thus
By Propositions 3.2 and 3.4, we obtain the main result in this section.
Theorem 3.5.
-
(1)
if ( and only if )for some distinct primes . -
(2)
if ( and only if )n is a multiple of the square of some prime.
-
Figure 3. The coloring of some zero-divisor graphs
Acknowledgements.
We would like to thank the referee for several valuable suggestions.
References
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