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 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
Boletín de la Sociedad Matemática Mexicana – Springer Journals
Published: May 31, 2018
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