Zariski-Like Topology on S-Quasi-Primary Ideals of a Commutative Ring

Abstract

Let $R$ be a commutative ring with nonzero identity and, $S\subseteq R$ be a multiplicatively closed subset. An ideal $P$ of $R$ is called an $S$ -quasi-primary ideal if $P\cap S=\varnothing$ and there exists an (fixed) $s\in S$ and whenever $ab\in P$ for $a,b\in R$ then either $sa\in \sqrt{P}$ or $sb\in \sqrt{P}\text{.}$ In this paper, we construct a topology on the set $Q{\text{Prim}}_S\left(R\right)$ of all $S$ -quasi-primary ideals of $R$ which is a generalization of the $S$ -prime spectrum of $R$ . Also, we investigate the relations between algebraic properties of $R$ and topological properties of $Q{\text{Prim}}_S\left(R\right)$ like compactness, connectedness and irreducibility.