@article { author = {Bijari, H. and Khashyarmanesh, K. and Fazaeli Moghim, H.}, title = {PRIMARY ZARISKI TOPOLOGY ON THE PRIMARY SPECTRUM OF A MODULE}, journal = {Journal of Algebraic Systems}, volume = {8}, number = {1}, pages = {53-68}, year = {2020}, publisher = {Shahrood University of Technology}, issn = {2345-5128}, eissn = {2345-511X}, doi = {10.22044/jas.2019.8320.1407}, 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}, url = {https://jas.shahroodut.ac.ir/article_1766.html}, eprint = {https://jas.shahroodut.ac.ir/article_1766_bb94c6f535b2d77ed688e10b285d39ea.pdf} }