Bol. Soc. Mat. Mex.
Nearly spectral spaces
· 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
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 asthesober, 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 notion has two natural generalizations: the ﬁrst in the context of
rings and the second in the context of lattices:
Ibeth Marcela Rubio Perilla
Mathematics Department, Universidad Nacional de Colombia, AK 30 45-03, Bogotá, Colombia