Skip to main content
SpringerLink
Log in

Patch-generated Frames and Projectable Hulls

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

This article considers coherent frame homomorphisms h : L → M between coherent frames, which induce an isomorphism between the boolen frames of polars, with M projectable, and such that M is generated by h(L) and certain complemented elements of M. This abstracts the passage from a semiprime commutative ring with identity to its projectable hull. The frame theoretic setting is investigated thoroughly, first without any assumptions beyond the Zermelo–Fraenkel axioms of set theory, and, subsequently, assuming that algebraic frames are spatial. The culmination of this effort is the result that the spectrum of d-elements of M is obtained from that of L by refining the given hull–kernel topology to the patch topology. The second part of the article relates the projectable hull to the (von Neumann) regular hull, in a variety of contexts, including that of f-rings. For a uniformly complete f-algebra A, it is shown that the maximal ℓ-ideals of A that are traces of real maximal ideals of the regular hull HA are precisely the almost P-points of the space of maximal ℓ-ideals of A.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson, F.W.: Lattice-ordered rings of quotients. Canad. J. Math. 17, 434–448 (1965)

    MATH  MathSciNet  Google Scholar 

  2. Aron, E.R., Hager, A.W.: Convex vector lattices and ℓ-algebras. Topology Appl. 121, 1–10 (1981)

    Article  MathSciNet  Google Scholar 

  3. Ball, R.N., Hager, A.W.: Characterization of epimorphisms in archimedean lattice-ordered groups and vector lattices. In: Glass, A.M.W., Holland, W.C. (eds.) Lattice-ordered Groups, Advances and Techniques. Kluwer, Dordrecht (1989)

    Google Scholar 

  4. Banaschewski, B.: Maximal rings of quotients of semisimple commutative rings. Arch. Math. XVI, 414–420 (1965)

    Article  MathSciNet  Google Scholar 

  5. Banaschewski, B.: Radical ideals and coherent frames. Comment. Math. Univ. Carolin. 37, 349–370 (1996)

    MATH  MathSciNet  Google Scholar 

  6. Banaschewski, B., Pultr, A.: Booleanization. Cahiers Topologie Géom. Differentielle Catég. XXXVII-1, 41–60 (1996)

    MathSciNet  Google Scholar 

  7. Bigard, A., Keimel, K., Wolfenstein, S.: Groupes et Anneaux Réticulés. In: Lecture Notes in Math, vol. 608. Springer, Berlin Heidelberg New York (1977)

    Google Scholar 

  8. Conrad, P.F., Martínez, J.: Complemented lattice-ordered groups. Indag. Math. (N.S.) 1(3), 281–298 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Darnel, M.R.: The Theory of Lattice-Ordered Groups. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 187. Marcel Dekker, New York (1995)

    Google Scholar 

  10. Fine, N., Gillman, L., Lambek, J.: Rings of quotients of rings of functions. Lecture Notes in Real Algebraic and Analytic Geometry, RAAG Network, Passau (2005)

  11. Gillman, L., Jerison, M.: Rings of Continuous Functions. In: Graduate Texts in Mathematics, vol. 43. Springer, Berlin Heidelberg New York (1976)

    Google Scholar 

  12. Hager A.W., Martínez, J.: Fraction dense algebras and spaces. Canad. J. Math. 45(5), 977–996 (1993)

    MATH  MathSciNet  Google Scholar 

  13. Hager, A.W., Martínez, J.: Hulls for various kinds of α-completeness in archimedean lattice-ordered groups. Order 16, 89–103 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Henriksen, M., Jerison, M.: The space of minimal prime ideals of a commutative ring. Trans. AMS 115, 110–130 (1976)

    Article  MathSciNet  Google Scholar 

  15. Henriksen, M., Johnson, D.G.: On the structure of a class of lattice-ordered algebras. Fund. Math. 50, 73–94 (1961)

    MATH  MathSciNet  Google Scholar 

  16. Hochster, M.: Prime ideal structure in commutative rings. Trans. AMS 142, 43–60 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  17. Huijsmans, C.B., de Pagter, B.: On z-ideals and d-ideals in Riesz spaces, I. Indag. Math. 42, 183–195 (1980)

    Google Scholar 

  18. Huijsmans, C.B., de Pagter, B.: On z-ideals and d-ideals in Riesz spaces, II. Indag. Math. 42, 391–408 (1980)

    Google Scholar 

  19. Huijsmans, C.B., de Pagter, B.: Maximal d-ideals in a Riesz space. Canad. J. Math. 35, 1010–1029 (1983)

    MATH  MathSciNet  Google Scholar 

  20. Johnstone, P.T.: Stone Spaces. Cambridge Studies in Adv. Math, vol. 3. Cambridge University Press, Cambridge (1982)

    Google Scholar 

  21. Lambek, J.: Lectures on Rings and Modules (3rd edn.) Chelsea, New York (1986)

    Google Scholar 

  22. Levy, R.: Almost P-spaces. Canad. J. Math. 29(2), 284–288 (1977)

    MATH  MathSciNet  Google Scholar 

  23. Martínez, J.: Archimedean lattices. Algebra Universalis 3(fasc. 2), 247–260 (1973)

    MATH  Google Scholar 

  24. Martínez, J.: The maximal ring of quotients of an f-ring. Algebra Universalis 33, 355–369 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  25. Martínez, J.: Polar functions, I: the summand-inducing hull of an archimedean lattice-ordered group. In: Martínez, J. (ed.), Ordered Algebraic Structures, Proc. 2001 Gainesville Conf. pp. 275–299 (2002)

  26. Martínez, J.: Unit and kernel systems in algebraic frames. Algebra Universalis 55, 13–43 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Martínez, J., Zenk, E.R.: When an algebraic frame is regular. Algebra Universalis 50, 231–257 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Martínez, J., Zenk, E.R.: Nuclear typings of frames vs spatial selectors. Appl. Categ. Structures 14, 35–61 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  29. Martínez, J., Zenk, E.R.: Regularity in algebraic frames. (submitted)

  30. Martínez, J., Zenk, E.R.: Epicompletion in frames with skeletal maps, II: Coherent normal archimedean frames. (work in progress)

  31. Monteiro, A.: L’Arithmétique des filtres et les espaces topologiques. Segundo Symposium de Matemática; Villavicencio (Mendoza), pp. 129–162 (1954)

  32. Raphael, R.M., Woods, R.G.: The epimorphic hull of C(X). Topology Appl. 105, 65–88 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  33. Roby, N.: Diverses caractérisations des épimorphismes. In: Les Épimorphismes d’Anneaux. Sémin. d’Alg. Comm. (P. Samuel, Dir), École Norm. Sup. de J. Filles; Paris (1968)

  34. Storrer, H.H.: Epimorphismen von kommutativen Ringen. Comment. Math. Helv. 43, 378–401 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  35. Utumi, Y.: On quotient rings. Osaka J. Math. 8, 1–18 (1956)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Wesleyan University, Middletown, CT, 06459, USA

    Anthony W. Hager

  2. Department of Mathematics, University of Florida, Box 118105, Gainesville, FL, 32611-8105, USA

    Jorge Martínez

Authors
  1. Anthony W. Hager
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jorge Martínez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jorge Martínez.

Additional information

For Bernhard Banaschewski, on the occasion of his 80th birthday.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hager, A.W., Martínez, J. Patch-generated Frames and Projectable Hulls. Appl Categor Struct 15, 49–80 (2007). https://doi.org/10.1007/s10485-007-9062-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-007-9062-y

Key words

Mathematics Subject Classifications (2000)

Search

Navigation