Research ArticleNo Access

Associate class graph of zero-divisors of a commutative ring

Gaohua Tang

School of Sciences, Beibu Gulf University, Qinzhou 535011, P. R. China

School of Mathematics and Statistics, Nanning Normal University, Nanning 530001, P. R. China

E-mail Address: tanggaohua@163.com

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Guangke Lin

Beihai Vocational College, Beihai 536000, P. R. China

E-mail Address: lgk1213@163.com

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Yansheng Wu

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, P. R. China

E-mail Address: wysasd@163.com

Corresponding author.

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https://doi.org/10.1142/S0219498820501558Cited by:1 (Source: Crossref)

Abstract

In this paper, we introduce the concept of the associate class graph of zero-divisors of a commutative ring $R$ $R$ , denoted by $Γ_{A} (R)$ $Γ_{A} (R)$ . Some properties of $Γ_{A} (R)$ $Γ_{A} (R)$ , including the diameter, the connectivity and the girth are investigated. Utilizing this graph, we present a new class of counterexamples of Beck’s conjecture on the chromatic number of the zero-divisor graph of a commutative ring.

Communicated by L. A. Bokut

Keywords:

AMSC: 13A99, 05C99

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