PRIMARY ZARISKI TOPOLOGY ON THE PRIMARY SPECTRUM OF A MODULE

Abstract

‎‎Let $R$ be a commutative ring with identity and let $M$ be an $R$-module‎. ‎We define the primary spectrum of $M$‎, ‎denoted by $\mathcal{PS}(M)$‎, ‎to be the set of all primary submodules $Q$ of $M$ such that $(\operatorname{rad}Q:M)=\sqrt{(Q:M)}$‎. ‎In this paper‎, ‎we topologize $\mathcal{PS}(M)$ with a topology having the Zariski topology on the prime spectrum $\operatorname{Spec}(M)$ as a subspace topology‎. ‎We investigate compactness and irreducibility of this topological space and provide some conditions under which $\mathcal{PS}(M)$ is a spectral space‎.

Keywords

• primary spectrum‎
• ‎primary zariski topology‎
• ‎primary submodule‎
• ‎prime ideal