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 <TEX>$T_1$</TEX>-subspace. We also study the zero-divisor graph <TEX>$\Gamma_I$</TEX>(R) with respect to the completely semiprime ideal I of N. We show that <TEX>${\Gamma}_{\mathbb{P}}$</TEX> (R), where <TEX>$\mathbb{P}$</TEX> 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 <TEX>${\Gamma}_{\mathbb{P}}$</TEX> (R).
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