PRIMARY ZARISKI TOPOLOGY ON THE PRIMARY SPECTRUM OF A MODULE

Document Type : Original Manuscript

Authors

1 Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159- 91775, Mashhad, Iran.

2 Department of Mathematics, University of Birjand, P.O. Box 97175-615, Birjand, Iran.

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

References

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Journal of Algebraic Systems
Volume 8, Issue 1
September 2020
Pages 53-68

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