An ideal-based zero-divisor graph of 2-primal near-rings
Bull. Korean Math. Soc. 2009 Vol. 46, No. 6, 1051-1060
Published online November 1, 2009
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
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