Algebra universalis

, Volume 74, Issue 3–4, pp 411–424 | Cite as

Covering maximal ideals with minimal primes

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Abstract

A ring is a UMP-ring if every maximal ideal in the ring is the union of the minimal prime ideals it contains. Banerjee, Ghosh and Henriksen have characterized Tychonoff spaces X for which C(X) is a UMP-ring. One of the characterizations is that every singleton of β X is what is called a nearly round subset. In this article, we define nearly round quotient maps, and use them to characterize completely regular frames L for which \({\mathcal{R} L}\) is a UMP-ring. All such frames are almost P-frames, and an Oz-frame is of this kind precisely when it is an almost P-frame. If L is perfectly normal (and hence if L is metrizable), then \({\mathcal{R} L}\) is a UMP-ring if and only if L is Boolean. If A is a UMP-ring which is a \({\mathbb{Q}}\) -algebra, then every ideal of A, when viewed as a ring in its own right, is a UMP-ring.

Keywords and phrases

frame ring of continuous real-valued functions on a frame f-ring maximal ideal minimal prime ideal UMP-ring 

2010 Mathematics Subject Classification

Primary: 06D22 Secondary: 54E17 13A15 18A40 
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References

  1. 1.
    Artico G., Marconi U.: On the compactness of the minimal spectrum. Rend. Sem. Mat. Univ. Padova 56, 79–84 (1976)MathSciNetMATHGoogle Scholar
  2. 2.
    Ball R.N., Walters-Wayland J.: C-and C*-quotients in pointfree topology. Dissertationes Math. (Rozprawy Mat.) 412, 62 (2002)MathSciNetMATHGoogle Scholar
  3. 3.
    Banaschewski, B.: The real numbers in pointfree topology. Textos de Matemática Série B, No. 12. Departamento de Matemática da Universidade de Coimbra (1997)Google Scholar
  4. 4.
    Banaschewski B.: On the function rings of pointfree topology. Kyungpook Math. J. 48, 195–206 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Banaschewski B., Dube T., Gilmour C., Walters-Wayland J.: OZ in pointfree topology. Quaest. Math. 32, 215–227 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Banaschewski B., Gilmour C.: Realcompactness and the cozero part of a frame. Appl. Categ. Structures 9, 395–417 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Banaschewski B., Hong S.S.: Completeness properties of function rings in pointfree topology. Comment. Math. Univ. Carolin. 44, 245–259 (2003)MathSciNetMATHGoogle Scholar
  8. 8.
    Banerjee B., Ghosh S.K., Henriksen M.: Unions of minimal prime ideals of rings of continuous functions on compact spaces. Algebra Universalis 62, 239–246 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Blair R.L.: Spaces in which special sets are z-embedded. Canad. J.Math. 28, 673–690 (1984)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Contessa M.: Ultraproducts of pm-rings and mp-rings. J. Pure Appl. Algebra 32, 11–20 (1984)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cornish W.H.: Normal lattices. J. Aust. Math. Soc. 14, 200–215 (1972)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    De Marco G., Orsatti A.: Commutative rings in which every prime ideal is contained in a unique maximal ideal. Proc. Amer. Math. Soc. 30, 459–466 (1971)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dube T.: Remote points and the like in pointfree topology. Acta Math. Hungar. 123, 203–222 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Dube T.: Some ring-theoretic properties of almost P-frames. Algebra Universalis 60, 145–162 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dube, T.: Concerning P-frames, essential P-frames, and strongly zero-dimensional frames Algebra Universalis 61, 115–138 (2009)Google Scholar
  16. 16.
    Dube T.: Some algebraic characterizations of F-frames. Algebra Universalis 62, 273–288 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gillman L., Jerison M.: Rings of Continuous Functions. Van Nostrand, Princeton (1960)CrossRefMATHGoogle Scholar
  18. 18.
    Ghosh S.K.: Intersections of minimal prime ideals in the rings of continuous functions. Comment. Math. Univ. Carolin. 47, 623–632 (2006)MathSciNetMATHGoogle Scholar
  19. 19.
    Gutiérrez García J., Kubiak T., Picado J.: Pointfree forms of Dowker’s and Michael’s insertion theorems. J. Pure Appl. Algebra 213, 98–108 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Henriksen M., Jerison M.: The space of minimal prime ideals of a commutative ring. Trans. Amer. Math. Soc. 115, 110–130 (1965)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Madden J., Vermeer J.: Lindelöf locales and realcompactness. Math. Proc. Cambridge Philos. Soc. 99, 473–480 (1986)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Matlis E.: The minimal prime spectrum of a reduced ring. Illinois J. Math. 27, 353–391 (1983)MathSciNetMATHGoogle Scholar
  23. 23.
    Picado J., Pultr A.: Frames and Locales: topology without points Frontiers in Mathematics. Springer, Basel (2012)CrossRefMATHGoogle Scholar
  24. 24.
    Rudd D.: On isomorphisms between ideals in rings of continuous functions. Trans. Amer. Math. Soc. 159, 335–353 (1971)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of South AfricaPretoriaSouth Africa

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