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

摘要

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 $Gamma_{mathbb{P}}(R).$