Skip to main content
SpringerLink
Log in

Model companion and model completion of theories of rings

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

Extending the language of rings to include predicates for Jacobson radical relations, we show that the theory of regular rings defined by Carson, Lipshitz and Saracino is the model completion of the theory of semisimple rings. Removing the requirement on the Jacobson radical (reduced to {0}), we prove that the theory of rings with no nilpotents does not admit a model companion relative to this augmented language.

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.

Similar content being viewed by others

References

  1. Carson A.: The model-completion of the theory of commutative regular rings. J. Algebra 27, 136–146 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chang C., Keisler H.: Model Theory. North-Holland, Amsterdam (1973)

    MATH  Google Scholar 

  3. Cherlin G.: Model Theoretic Algebra; Selected Topics. Lecture Notes in Mathematics, vol. 521. Springer, Berlin (1976)

    Google Scholar 

  4. Hodges W.: Model Theory. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  5. Lipshitz L., Saracino D.: The model companion of the theory of commutative rings without nilpotent elements. Proc. Am. Math. Soc. 38(2), 381–387 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  6. Macintyre A.: Model-completeness for sheaves of structures. Fund. Math. 81, 73–89 (1973)

    MathSciNet  Google Scholar 

  7. Macintyre A., McKenna K., van den Dries L.: Elimination of quantifiers in algebraic structures. Adv. Math. 47, 76–87 (1983)

    Article  Google Scholar 

  8. Macintyre A., van den Dries L.: The logic of Rumely’s local-global principle. J. Reine Angew. Math. 407, 33–56 (1990)

    MathSciNet  MATH  Google Scholar 

  9. Prestel A.: Einführung in die Mathematische Logik und Modelltheorie. Vieweg, Braunschweig-Wiesbaden (1986)

    MATH  Google Scholar 

  10. Prestel A., Schmid J.: Existentially closed domains with radical relations. J. Reine Angew. Math. 407, 178–201 (1990)

    MathSciNet  MATH  Google Scholar 

  11. Sureson C.: A generalization of von Neumann regularity. Ann. Pure Appl. Logic 135(1–3), 210–242 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Sureson C.: A valuation ring analogue of von Neumann regularity. Ann. Pure Appl. Logic 145(2), 204–222 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. van den Dries L.: Elimination theory for the ring of algebraic integers. J. Reine Angew. Math. 407, 189–205 (1988)

    Google Scholar 

  14. Weispfenning V.: Model completeness and elimination of quantifiers for subdirect products of structures. J. Algebra 36, 252–277 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  15. Weispfenning V.: On the model theory of valued regular rings. J. Symb. Logic 42(3), 477 (1977)

    MathSciNet  Google Scholar 

  16. Weispfenning, V.: Valuation rings and boolean products. In: Proc. Conf. F.N.R.S., Bruxelles (1982)

Download references

Author information

Authors and Affiliations

  1. Equipe de Logique Mathématique, Université Paris 7, Paris, France

    Claude Sureson

Authors
  1. Claude Sureson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claude Sureson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sureson, C. Model companion and model completion of theories of rings. Arch. Math. Logic 48, 403–420 (2009). https://doi.org/10.1007/s00153-009-0129-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-009-0129-3

Keywords

Mathematics Subject Classification (2000)

Search

Navigation