Algebraic properties of rings of continuous functions

M. Mulero

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 1, page 55-66
  • ISSN: 0016-2736

Abstract

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This paper is devoted to the study of algebraic properties of rings of continuous functions. Our aim is to show that these rings, even if they are highly non-noetherian, have properties quite similar to the elementary properties of noetherian rings: we give going-up and going-down theorems, a characterization of z-ideals and of primary ideals having as radical a maximal ideal and a flatness criterion which is entirely analogous to the one for modules over principal ideal domains.

How to cite

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Mulero, M.. "Algebraic properties of rings of continuous functions." Fundamenta Mathematicae 149.1 (1996): 55-66. <http://eudml.org/doc/212108>.

@article{Mulero1996,
abstract = {This paper is devoted to the study of algebraic properties of rings of continuous functions. Our aim is to show that these rings, even if they are highly non-noetherian, have properties quite similar to the elementary properties of noetherian rings: we give going-up and going-down theorems, a characterization of z-ideals and of primary ideals having as radical a maximal ideal and a flatness criterion which is entirely analogous to the one for modules over principal ideal domains.},
author = {Mulero, M.},
journal = {Fundamenta Mathematicae},
keywords = {rings of continuous functions; going-up and going-down theorems; z-ideals; primary ideals; flat modules; primary ideal; flat module},
language = {eng},
number = {1},
pages = {55-66},
title = {Algebraic properties of rings of continuous functions},
url = {http://eudml.org/doc/212108},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Mulero, M.
TI - Algebraic properties of rings of continuous functions
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 1
SP - 55
EP - 66
AB - This paper is devoted to the study of algebraic properties of rings of continuous functions. Our aim is to show that these rings, even if they are highly non-noetherian, have properties quite similar to the elementary properties of noetherian rings: we give going-up and going-down theorems, a characterization of z-ideals and of primary ideals having as radical a maximal ideal and a flatness criterion which is entirely analogous to the one for modules over principal ideal domains.
LA - eng
KW - rings of continuous functions; going-up and going-down theorems; z-ideals; primary ideals; flat modules; primary ideal; flat module
UR - http://eudml.org/doc/212108
ER -

References

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  1. [1] M. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, 1969. 
  2. [2] R. Bkouche, Couples spectraux et faisceaux associés. Applications aux anneaux de fonctions, Bull. Soc. Math. France 98 (1970), 253-295. Zbl0201.37204
  3. [3] N. Bourbaki, Algèbre commutative, Ch. 1 and 2, Hermann, 1961. 
  4. [4] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, 1976. 
  5. [5] T. Isiwata, Mappings and spaces, Pacific J. Math. 20 (1967), 455-480. Zbl0149.40501
  6. [6] C. W. Khols, Prime ideals in rings of continuous functions II, Duke Math. J. 2 (1958), 447-458. 
  7. [7] W. S. Massey, Algebraic Topology: An Introduction, Springer, 1967. 
  8. [8] H. Matsumura, Commutative Algebra, 2nd ed., Benjamin, 1980. 
  9. [9] M. A. Mulero Díaz, Revestimientos finitos y álgebras de funciones continuas, Ph.D. Thesis, Univ. de Extremadura, 1992. 
  10. [10] J. Muñoz Díaz, Caracterización de las álgebras diferenciales y síntesis espectral para módulos sobre tales álgebras, Collect. Math. 23 (1972), 17-83. 
  11. [11] C. W. Neville, Flat C(X)-modules and F-spaces, Math. Proc. Cambridge Philos. Soc. 106 (1989), 237-244. Zbl0780.54016

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