Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces

Othman Echi

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 2, page 489-507
  • ISSN: 0392-4041

Abstract

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In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if g : Y → X is a quasi-homeomorphism, Z a sober space and f : Y → Z a continuous map, then there exists a unique continuous map F : X → Z such that F ∘ g = f . Let X be a T 0 -space, q : X → s X the injection of X onto its sobrification X s . It is shown, here, that q ⁢ Gold ⁢ X = Gold ⁢ X s , where Gold ⁢ X is the set of all locally closed points of X . Some applications are also indicated. The Jacobson prime spectrum of a commutative ring R is the set of all prime ideals of R which are intersections of some maximal ideals of R . One of our main results is a surprising answer to the problem of ordered disjoint union of jacspectral sets (ordered sets which are isomorphic to the Jacobson prime spectrum of some ring): Let { ( X λ , ≤ λ ) : λ ∈ Λ } be a collection of ordered disjoint sets and X = ⋃ λ ∈ Λ X λ . Partially order X by declaring x ≤ y to mean that there exists λ ∈ Λ such that x , y ∈ X λ and x ≤ λ y . Then the following statements are equivalent: (i) ( X , ≤ ) is jacspectral. (ii) ( X λ , ≤ λ ) is jacspectral, for each λ ∈ Λ .

How to cite

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Echi, Othman. "Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces." Bollettino dell'Unione Matematica Italiana 6-B.2 (2003): 489-507. <http://eudml.org/doc/196084>.

@article{Echi2003,
abstract = {In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if $g \colon Y \to X$ is a quasi-homeomorphism, $Z$ a sober space and $f \colon Y \to Z$ a continuous map, then there exists a unique continuous map $F \colon X \to Z$ such that $F \circ g =f$. Let $X$ be a $T_\{0\}$-space, $q \colon X \to^\{s\} X$ the injection of $X$ onto its sobrification $^\{s\}X$. It is shown, here, that $q(\text\{Gold\}(X))=\text\{Gold\}(\sideset\{^\{s\}\}\{\}\{\operatorname\{X\}\})$, where $\text\{Gold\}(X)$ is the set of all locally closed points of $X$. Some applications are also indicated. The Jacobson prime spectrum of a commutative ring $R$ is the set of all prime ideals of $R$ which are intersections of some maximal ideals of $R$. One of our main results is a surprising answer to the problem of ordered disjoint union of jacspectral sets (ordered sets which are isomorphic to the Jacobson prime spectrum of some ring): Let $\\{(X_\{\lambda\}, \leq_\{\lambda\}) \, : \, \lambda\in\Lambda \\}$ be a collection of ordered disjoint sets and $X=\bigcup_\{\lambda\in\Lambda\} X_\{\lambda\}$. Partially order $X$ by declaring $x\leq y$ to mean that there exists $\lambda\in\Lambda$ such that $x$, $y\in X_\{\lambda\}$ and $x\leq_\{\lambda\} y$. Then the following statements are equivalent: (i) $(X, \leq)$ is jacspectral. (ii) $(X_\{\lambda\}, \leq_\{\lambda\})$ is jacspectral, for each $\lambda\in\Lambda$.},
author = {Echi, Othman},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {489-507},
publisher = {Unione Matematica Italiana},
title = {Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces},
url = {http://eudml.org/doc/196084},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - Echi, Othman
TI - Quasi-homeomorphisms, Goldspectral spaces and Jacspectral spaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/6//
PB - Unione Matematica Italiana
VL - 6-B
IS - 2
SP - 489
EP - 507
AB - In this paper, we deal with the study of quasi-homeomorphisms, the Goldman prime spectrum and the Jacobson prime spectrum of a commutative ring. We prove that, if $g \colon Y \to X$ is a quasi-homeomorphism, $Z$ a sober space and $f \colon Y \to Z$ a continuous map, then there exists a unique continuous map $F \colon X \to Z$ such that $F \circ g =f$. Let $X$ be a $T_{0}$-space, $q \colon X \to^{s} X$ the injection of $X$ onto its sobrification $^{s}X$. It is shown, here, that $q(\text{Gold}(X))=\text{Gold}(\sideset{^{s}}{}{\operatorname{X}})$, where $\text{Gold}(X)$ is the set of all locally closed points of $X$. Some applications are also indicated. The Jacobson prime spectrum of a commutative ring $R$ is the set of all prime ideals of $R$ which are intersections of some maximal ideals of $R$. One of our main results is a surprising answer to the problem of ordered disjoint union of jacspectral sets (ordered sets which are isomorphic to the Jacobson prime spectrum of some ring): Let $\{(X_{\lambda}, \leq_{\lambda}) \, : \, \lambda\in\Lambda \}$ be a collection of ordered disjoint sets and $X=\bigcup_{\lambda\in\Lambda} X_{\lambda}$. Partially order $X$ by declaring $x\leq y$ to mean that there exists $\lambda\in\Lambda$ such that $x$, $y\in X_{\lambda}$ and $x\leq_{\lambda} y$. Then the following statements are equivalent: (i) $(X, \leq)$ is jacspectral. (ii) $(X_{\lambda}, \leq_{\lambda})$ is jacspectral, for each $\lambda\in\Lambda$.
LA - eng
UR - http://eudml.org/doc/196084
ER -

References

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  1. ARTIN, E.- TATE, J. T., A note on finite ring extension, J. Math. Soc. Japan, 3 (1951), 74-77. Zbl0043.26701MR44509
  2. AULL, C. E.- THRON, W. J., Separation axioms between T 0 and T 1 , Indag. Math., 24 (1962), 26-37. Zbl0108.35402MR138082
  3. BOUACIDA, E.- ECHI, O.- SALHI, E., Topologies associées à une relation binaire et relation binaire spectrale, Boll. Un. Mat. Ital. (7), 10-B (1996), 417-439. Zbl0865.54032MR1397356
  4. BOUACIDA, E.- ECHI, O.- SALHI, E., Foliations, spectral topology and special morphisms, Lect. Notes Pure Appl. Math. (Dekker), 205 (1999), 111-132. Zbl0957.57017MR1767454
  5. BOUACIDA, E.- ECHI, O.- SALHI, E., Feuilletages et topologie spectrale, J. Math. Soc. Japan, 52 (2000), 447-464. Zbl0965.57025MR1742794
  6. BOUACIDA, E.- ECHI, O.- SALHI, E., Goldman points and Goldman topology, Submitted for publication. 
  7. BOUVIER, A.- FONTANA, M., Une classe d'espaces spectraux de dimension ≤ 1 : les espaces principaux, Bull. Sc. Math. 2e série, 105 (1981), 159-167. Zbl0459.13001MR618875
  8. CONTE, A., Proprietà di separazione della topologia di Zariski di uno schema, Ist. Lombardo Accad. Sci. Lett. Rend., Ser. A, 106 (1972), 79-111. Zbl0254.14002MR349670
  9. DOBBS, D. E.- FONTANA, M.- PAPICK, I. J., On certain distinguished spectral sets, Ann. Mat. Pura Appl. (4), 128 (1980), 227-240. Zbl0472.54021MR640784
  10. ECHI, O., A topological characterization of the Goldman prime spectrum of commutative ring, Comm. Algebra, 28 (5) (2000), 2329-2337. Zbl0976.13014MR1757465
  11. FONTANA, M., Quelques nouveaux résultats sur une classe d'espaces spectraux, Rend. Acad. Naz. Lincei, Serie, VIII, 67 (1979), 157-161. Zbl0461.13002MR622786
  12. FONTANA, M.- MAROSCIA, P., Sur les anneaux de Goldman, Boll. Un. Mat. Ital., 13-B (1976), 743-759. Zbl0351.13002MR463154
  13. GROTHENDIECK, A.- DIEUDONNE, J., Eléments de géométrie algébrique, Springer Verlag (1971). Zbl0203.23301
  14. GOLDMAN, O., Hilbert rings and the Hilbert Nullstallensatz, Math. Z., 54 (1951), 136-140. Zbl0042.26401MR44510
  15. HOCHSTER, M., Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142 (1969), 43-60. Zbl0184.29401MR251026
  16. HOCHSTER, M., The minimal prime spectrum of a commutative ring, Canad. J. Math., 23 (1971), 749-758. Zbl0216.19304MR292805
  17. JOYAL, A., Spectral spaces and distributive lattices, Notices Amer. Math. Soc., 18 (1971), 393-394. 
  18. JOYAL, A., Spectral spaces II, Notices Amer. Math. Soc., 18 (1971), 618. 
  19. KAPLANSKY, I., Commutative rings (revised edition), The University of Chicago Press, Chicago (1974). Zbl0296.13001MR345945
  20. KRULL, W., Jacobsonsche Ring, Hilbertscher Nullstellensatz Dimensionnentheorie, Math. Z., 54 (1951), 354-387. Zbl0043.03802MR47622
  21. LEWIS, W. J., The spectrum of a ring as a partially ordered set, J. Algebra, 25 (1973), 419-434. Zbl0266.13010MR314811
  22. LEWIS, W. J.- OHM, J., The ordering of Spec ⁢ R , Canad. J. Math., 28 (1976), 820-835. Zbl0313.13003MR409428
  23. PICAVET, G., Sur les anneaux commutatifs dont tout idéal premier est de Goldman, C. R. Acad. Sci. Paris Sér A, 280 (1975), 1719-1721. Zbl0328.13001MR469900
  24. PICAVET, G., Autour des idéaux premiers de Goldman d'un anneau commutatif, Ann. Sc. Univ. Clermont Math., 57 (1975), 73-90. Zbl0317.13002MR392967
  25. PRIESTLEY, H. A., Spectral sets, J. Pure. Appl. Algebra, 94 (1994), 101-114. Zbl0807.06001MR1277526

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