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 PS(M), to be the set of all primary submodules Q of M such that (rad Q : M) = √ (Q : M). In this paper, we topologize PS(M) with a topology having the Zariski topology on the prime spectrum Spec(M) as a subspace topology. We investigate compactness and irreducibility of this topological space and provide some conditions under which PS(M) is a spectral space.