Characterization of $2$-Primal Near-Rings

Abstract:

      In 1999, Kim and Kwak asked one question that ``Is a ring $R$ $2$-primal if $O_{P}\subseteq P$ for each $P\in m{\rm Spec}(R)?"$. In this paper, we prove that if $O_{P}$ has the IFP for each $P \in m{\rm Spec}(N)$, then $O_{P} \subseteq P$ for each $P \in m{\rm Spec}(N)$ if and only if $N$ is a $2$-primal near-ring and also we give characterization of 2-primal near-rings by using its minimal $0$-prime ideals.