Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2009; 46(6): 1051-1060

Printed November 1, 2009

https://doi.org/10.4134/BKMS.2009.46.6.1051

Copyright © The Korean Mathematical Society.

An ideal-based zero-divisor graph of 2-primal near-rings

Patchirajulu Dheena and Balasubramanian Elavarasan

Annamalai University and Annamalai University

Abstract

In this paper, we give topological properties of collection of prime ideals in 2-primal near-rings. We show that Spec$(N),$ the spectrum of prime ideals, is a compact space, and Max$(N)$, the maximal ideals of $N,$ forms a compact $T_1$-subspace. We also study the zero-divisor graph $\Gamma_{I}(R)$ with respect to the completely semiprime ideal $I$ of $N.$ We show that $\Gamma_{\mathbb{P}}(R),$ where $\mathbb{P}$ is a prime radical of $N,$ is a connected graph with diameter less than or equal to 3. We characterize all cycles in the graph $\Gamma_{\mathbb{P}}(R).$

Keywords: graph, prime ideal, 2-primal, Zariski topology and cycle

MSC numbers: 16Y30, 13A99