Nearly spectral spaces

Nearly spectral spaces Bol. Soc. Mat. Mex. https://doi.org/10.1007/s40590-018-0206-x ORIGINAL ARTICLE 1 1 Lorenzo Acosta · Ibeth Marcela Rubio Perilla Received: 22 June 2017 / Accepted: 18 May 2018 © Sociedad Matemática Mexicana 2018 Abstract We study some natural generalizations of spectral spaces in the contexts of commutative rings and distributive lattices. We obtain a topological characterization for the spectra of commutative (not necessarily unitary) rings and we find spectral ver- sions for the up-spectral and down-spectral spaces. We show that the duality between distributive lattices and Balbes–Dwinger spaces is the co-equivalence associated with a pair of contravariant right adjoint functors between suitable categories. Keywords Spectral space · Down-spectral space · Up-spectral space · Stone duality · Prime spectrum · Distributive lattice · Commutative ring Mathematics Subject Classification 54H10 · 54F65 · 54D35 1 Introduction A spectral space is a topological space that is homeomorphic to the prime spectrum of a commutative unitary ring. This type of spaces was topologically characterized by Hochster [8]asthe sober, coherent and compact spaces. On the other hand, it is known that a topological space is a spectral space if and only if it is homeomorphic to the prime spectrum of a distributive bounded lattice [1,10]. Therefore, this http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Boletín de la Sociedad Matemática Mexicana Springer Journals

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Sociedad Matemática Mexicana
Subject
Mathematics; Mathematics, general
ISSN
1405-213X
eISSN
2296-4495
D.O.I.
10.1007/s40590-018-0206-x
Publisher site
See Article on Publisher Site

Abstract

Bol. Soc. Mat. Mex. https://doi.org/10.1007/s40590-018-0206-x ORIGINAL ARTICLE 1 1 Lorenzo Acosta · Ibeth Marcela Rubio Perilla Received: 22 June 2017 / Accepted: 18 May 2018 © Sociedad Matemática Mexicana 2018 Abstract We study some natural generalizations of spectral spaces in the contexts of commutative rings and distributive lattices. We obtain a topological characterization for the spectra of commutative (not necessarily unitary) rings and we find spectral ver- sions for the up-spectral and down-spectral spaces. We show that the duality between distributive lattices and Balbes–Dwinger spaces is the co-equivalence associated with a pair of contravariant right adjoint functors between suitable categories. Keywords Spectral space · Down-spectral space · Up-spectral space · Stone duality · Prime spectrum · Distributive lattice · Commutative ring Mathematics Subject Classification 54H10 · 54F65 · 54D35 1 Introduction A spectral space is a topological space that is homeomorphic to the prime spectrum of a commutative unitary ring. This type of spaces was topologically characterized by Hochster [8]asthe sober, coherent and compact spaces. On the other hand, it is known that a topological space is a spectral space if and only if it is homeomorphic to the prime spectrum of a distributive bounded lattice [1,10]. Therefore, this

Journal

Boletín de la Sociedad Matemática MexicanaSpringer Journals

Published: May 31, 2018

References

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