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Primary Zariski topology on the primary spectrum of a module. (English) Zbl 1468.13022

Summary: 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.

MSC:

13C13 Other special types of modules and ideals in commutative rings
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