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Second Spectrum of Modules and Spectral Spaces

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Abstract

Let R be a commutative ring with identity and \(\hbox {Spec}^{s}(M)\) denote the set all second submodules of an R-module M. In this paper, we investigate various properties of \(\hbox {Spec}^{s}(M)\) with respect to different topologies. We investigate the dual Zariski topology from the point of view of separation axioms, spectral spaces and combinatorial dimension. We establish conditions for \(\hbox {Spec}^{s}(M)\) to be a spectral space with respect to quasi-Zariski topology and second classical Zariski topology. We also present some conditions under which a module is cotop.

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Acknowledgements

The authors would like to thank the Scientific Technological Research Council of Turkey (TUBITAK) for funding this work through the project 114F381. The M. Alkan is supported by the Scientific Research Project Administration of Akdeniz University.

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Correspondence to Seçil Çeken.

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Communicated by Siamak Yassemi.

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Çeken, S., Alkan, M. Second Spectrum of Modules and Spectral Spaces. Bull. Malays. Math. Sci. Soc. 42, 153–169 (2019). https://doi.org/10.1007/s40840-017-0473-0

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  • DOI: https://doi.org/10.1007/s40840-017-0473-0

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