[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.14317/jami.2022.729

THE ZERO-DIVISOR GRAPHS OF ℤ(+)ℤ_n AND (ℤ(+)ℤ_n)[X]]

PARK, MIN JI (Department of Mathematics, College of Life Science and Nano Technology, Hannam University)
JEONG, JONG WON (School of Mathematics, Kyungpook National University)
LIM, JUNG WOOK (Department of Mathematics, College of Natural Sciences, Kyungpook National University)
BAE, JIN WON (Department of Mathematics, College of Natural Sciences, Kyungpook National University)

Publication Information

Journal of applied mathematics & informatics / v.40, no.3_4, 2022 , pp. 729-740 More about this Journal

Abstract

Let ℤ be the ring of integers and let ℤ_n be the ring of integers modulo n. Let ℤ(+)ℤ_n be the idealization of ℤ_n in ℤ and let (ℤ(+)ℤ_n)[X]] be either (ℤ(+)ℤ_n)[X] or (ℤ(+)ℤ_n)[[X]]. In this article, we study the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]]. More precisely, we completely characterize the diameter and the girth of the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]]. We also calculate the chromatic number of the zero-divisor graphs of ℤ(+)ℤ_n and (ℤ(+)ℤ_n)[X]].

Keywords

${\Gamma}(\mathbb{Z}(+)\mathbb{Z}_n)$ ; ${\Gamma}((\mathbb{Z}(+)\mathbb{Z}_n)[X]])$ ; diameter; girth; clique; chromatic number;

Citations & Related Records

Times Cited By KSCI : 2 (Citation Analysis)

Reference
Cited By KSCI

1	S.B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra 30 (2002), 3533-3558. DOI
2	D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434-447. DOI
3	M. Axtell and J. Stickles, Zero-divisor graphs of idealizations, J. Pure Appl. Algebra 204 (2006), 235-243. DOI
4	J.A. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York and Basel, 1988.
5	D.B. West, Introduction to Graph Theory, 2nd ed., Prentice-Hall, Upper Saddle River, NJ, 2001.
6	D.F. Anderson, M.C. Axtell, and J.A. Stickles, Jr, Zero-divisor graphs in commutative rings, in: M. Fontana et al. (Eds), Commutative Algebra: Noetherian and Non-Noetherian Perspectives, Springer, New York, 2011, pp. 23-45.
7	D.D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 (1993), 500-514. DOI
8	D.E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math. Soc. 27 (1971), 427-433. DOI
9	J.W. Lim and D.Y. Oh, S-Noetherian properties on amalgamated algebras along an ideal, J. Pure Appl. Algebra 218 (2014), 1075-1080. DOI
10	N.H. McCoy, Remarks on divisors of zero, Amer. Math. Monthly 49 (1942), 286-295. DOI
11	M.J. Park, E.S. Kim, and J.W. Lim, The zero-divisor graph of ℤ_n[X]], Kyungpook Math. J. 60 (2020), 723-729. DOI
12	S.J. Pi, S.H. Kim, and J.W. Lim, The zero-divisor graph of the ring of integers modulo n, Kyungpook Math. J. 59 (2019), 591-601. DOI
13	D.D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 (2009), 3-56. DOI
14	I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208-226. DOI

KSCI

THE ZERO-DIVISOR GRAPHS OF ℤ(+)ℤn AND (ℤ(+)ℤn)[X]]

THE ZERO-DIVISOR GRAPHS OF ℤ(+)ℤ_n AND (ℤ(+)ℤ_n)[X]]