Toroidal zero-divisor graphs of decomposable commutative rings without identity
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ORIGINAL ARTICLE
Toroidal zero-divisor graphs of decomposable commutative rings without identity G. Kalaimurugan1 • P. Vignesh1 • T. Tamizh Chelvam2 Received: 16 August 2019 / Accepted: 26 February 2020 Ó Sociedad Matemática Mexicana 2020
Abstract Let R be a commutative ring without identity. The zero-divisor graph of R, denoted by CðRÞ; is a graph with vertex set ZðRÞnf0g; which is the set of all non-zero zerodivisor elements of R and two vertices x and y are adjacent if and only if xy ¼ 0: In this paper, we characterize (up to isomorphism) all finite decomposable commutative rings without identity whose zero-divisor graphs are toroidal. Keywords Commutative rings Nilpotent rings Decomposable rings Zero-divisor graph Genus
Mathematics Subject Classification 05C10 05C25 13M05
1 Introduction Throughout this paper, R denotes a commutative ring without identity. Let ZðRÞ; NilðRÞ; and annR ðSÞ denotes the set of all zero-divisors of R, the set of all nilpotent elements of R and the set of all annihilators of a subset S of R, respectively, and let R ¼ Rnf0g: Several authors [1, 2, 20–22] studied about graphs from algebraic structures and in particular graphs from rings. The study of graphs from commutative rings was initiated by Beck [11]. Later, Anderson and Livingston [5] defined the zero& T. Tamizh Chelvam [email protected]; [email protected] G. Kalaimurugan [email protected] P. Vignesh [email protected] 1
Department of Mathematics, Thiruvalluvar University, Vellore, Tamil Nadu 632 115, India
2
Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu 627 012, India
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divisor graph CðRÞ of a commutative ring R with identity. Actually, CðRÞ is the simple undirected graph with vertex set ZðRÞ and two distinct vertices x and y are adjacent if and only if xy ¼ 0: For a quick summary on zero-divisor graphs, one can refer the survey article [4]. Recently, Anderson and Weber [6] obtained some properties of zero-divisor graphs from finite commutative rings without identity. The concept of genus characterization of graphs has wide number of applications, especially in DNA strand and road map navigation. For finite commutative rings with identity, the planarity of zero-divisor graphs is studied in [3, 12] and the higher genus characterizations are described in [13, 14, 23, 27]. There are several studies regarding zero-divisor graphs from finite commutative rings without identity (see [8, 19, 24]). The planar characterization of zero-divisor graphs associated from nilpotent finite rings are obtained by Kuzmina and Maltsev in [17] and Kuzmina extended the characterization for finite nonnilpotent rings in [18]. Kalaimurugan et al.[16, Section 4] characterized all finite commutative rings of cubefree order whose zero-divisor graphs are toroidal. In this paper, now we obtain the classification of all finite decomposable commutative rings without identity whose zero-divisor graphs are toroidal. Let G ¼ ðV; EÞ be a simple undirected graph
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