Article
Kyungpook Mathematical Journal 2020; 60(4): 723729
Published online December 31, 2020
Copyright © Kyungpook Mathematical Journal.
The Zerodivisor Graph of ${\mathbb{Z}}_{n}[X]]$
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
email : mjpark5764@gmail.com
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566, Republic of Korea
email : eskim@knu.ac.kr
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566, Republic of Korea
email : 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 Zerodivisor Graph of a Commutative Ring
Let
In [6], Beck first introduced the concept of the zerodivisor 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 zerodivisor graph of a commutative ring, the readers can refer to a survey article [1].
Let
2. The Diameter and the Girth of
$\Gamma ({\mathbb{Z}}_{n}[X]\text{}])$
$\Gamma ({\mathbb{Z}}_{n}[X]\text{}])$
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
$f\in \text{Z}(R[X])$ , then there exists a nonzero elementr ∈ R such thatrf=0 . 
(2) If
R is a Noetherian ring and$f\in \text{Z}(R[\text{}[X]\text{}])$ , then there exists a nonzero elementr ∈ R such thatrf=0 .
Let
Theorem 2.2.
The following statements hold.

(1)
$\text{diam}(\Gamma ({\mathbb{Z}}_{n}[X]\text{}]))=1$ if (and only if )n=p^{2} for some primep . 
(2)
$\text{diam}(\Gamma ({\mathbb{Z}}_{n}[X]\text{}]))=2$ if (and only if )n=p^{r} for some primep and some integerr ≥ 3 , orn=pq for some distinct primesp andq . 
(3)
$\text{diam}(\Gamma ({\mathbb{Z}}_{n}[X]\text{}]))=3$ 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 zerodivisor 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)
$\text{g}(\Gamma ({\mathbb{Z}}_{n}[X]\text{}]))=3$ if (and only if ) each of the following conditions holds.
(a)
n=p^{r} for some primep and integerr ≥ 2 . 
(b)
n=pqr for some distinct primesp, q and integerr ≥ 2 .


(2)
$\text{g}(\Gamma ({\mathbb{Z}}_{n}[X]\text{}]))=4$ if (and only if )n=pq for some distinct primesp andq .

Figure 2. The girth of some zerodivisor graphs
3. The Chromatic Number of
$\Gamma ({\mathbb{Z}}_{n}[X]\text{}])$
$\Gamma ({\mathbb{Z}}_{n}[X]\text{}])$
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)
$\chi (\Gamma ({\mathbb{Z}}_{n}[X]\text{}]))=r$ if (and only if )$n={p}_{1}\cdots {p}_{r}$ for some distinct primes${p}_{1},\dots ,{p}_{r}$ . 
(2)
$\chi (\Gamma ({\mathbb{Z}}_{n}[X]\text{}]))=\infty $ if (and only if )n is a multiple of the square of some prime.

Figure 3. The coloring of some zerodivisor graphs
Acknowledgements.
We would like to thank the referee for several valuable suggestions.
References
 D. F. Anderson, M. C. Axtell, and J. A. Stickles, Jr, Zerodivisor graphs in commutative rings, Commutative Algebra: Noetherian and NonNoetherian Perspectives, 2345, Springer, New York, 2011.
 D. F. Anderson and P. S. Livingston, The zerodivisor graph of a commutative ring, J. Algebra, 217(1999), 434447.
 D. D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra, 159(1993), 500514.
 M. F. Atiyah and I. G. MacDonald, Introduction to commutative algebra, AddisonWesley Series in Math., Westview Press, 1969.
 M. Axtell, J. Coykendall, and J. Stickles, Zerodivisor graphs of polynomials and power series over commutative rings, Comm. Algebra, 33(2005), 20432050.
 I. Beck, Coloring of commutative rings, J. Algebra, 116(1988), 208226.
 D. E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc., 27(1971), 427433.
 S. Mulay, Cycles and symmetries of zerodivisors, Comm. Algebra, 30(2002), 35333558.
 S. J. Pi, S. H. Kim, and J. W. Lim, The zerodivisor graph of the ring of integers modulo n, Kyungpook Math. J., 59(2019), 591601.