Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Logic, Language, Information, and Computation

WoLLIC 2019: Logic, Language, Information, and Computation pp 533–549Cite as

An Algorithmic Approach to the Existence of Ideal Objects in Commutative Algebra

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11541))

Abstract

The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn’s lemma, and thus pose a challenge from a computational point of view. Giving a constructive meaning to ideal objects is a problem which dates back to Hilbert’s program, and today is still a central theme in the area of dynamical algebra, which focuses on the elimination of ideal objects via syntactic methods. In this paper, we take an alternative approach based on Kreisel’s no counterexample interpretation and sequential algorithms. We first give a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm. We then carry out a concrete case study, in which we give an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical.

The first, second and third author were supported by the German Science Foundation (DFG Project KO 1737/6-1); by the John Templeton Foundation (ID 60842) and by a Marie Skłodowska-Curie fellowship of the Istituto Nazionale di Alta Matematica, respectively. The opinions expressed in this paper are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The second author has contributed to a universal conservation criterion [31,32,33] that includes many of the those cases [30, 36, 39].

References

  1. Aschieri, F., Berardi, S.: Interactive learning-based realizability for Heyting arithmetic with EM1. Log. Methods Comput. Sci. 6(3), 1–22 (2010)

    Google Scholar 

  2. Atiyah, M., Macdonald, I.: Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Boston (1969)

    MATH  Google Scholar 

  3. Berger, U., Lawrence, A., Forsberg, F., Seisenberger, M.: Extracting verified decision procedures: DPLL and resolution. Log. Methods Comput. Sci. 11(1:6), 1–18 (2015)

    MathSciNet  MATH  Google Scholar 

  4. Berger, U., Miyamoto, K., Schwichtenberg, H., Seisenberger, M.: Minlog - a tool for program extraction supporting algebras and coalgebras. In: Corradini, A., Klin, B., Cîrstea, C. (eds.) CALCO 2011. LNCS, vol. 6859, pp. 393–399. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22944-2_29

    Chapter  MATH  Google Scholar 

  5. Cederquist, J., Coquand, T.: Entailment relations and distributive lattices. In: Buss, S.R., Hájek, P., Pudlák, P. (eds.) Logic Colloquium 1998, Proceedings of the Annual European Summer Meeting of the Association for Symbolic Logic. Lect. Notes Logic, Prague, Czech Republic, 9–15 August 1998, vol. 13, pp. 127–139. A. K. Peters, Natick (2000)

    Google Scholar 

  6. Cederquist, J., Negri, S.: A constructive proof of the Heine-Borel covering theorem for formal reals. In: Berardi, S., Coppo, M. (eds.) TYPES 1995. LNCS, vol. 1158, pp. 62–75. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61780-9_62

    Chapter  MATH  Google Scholar 

  7. Coste, M., Lombardi, H., Roy, M.F.: Dynamical method in algebra: Effective Nullstellensätze. Ann. Pure Appl. Logic 111(3), 203–256 (2001)

    Article  MathSciNet  Google Scholar 

  8. Gödel, K.: Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica 12, 280–287 (1958)

    Article  MathSciNet  Google Scholar 

  9. Gurevich, Y.: Sequential abstract-state machines capture sequential algorithms. ACM Trans. Comput. Log. (TOCL) 1, 77–111 (2000)

    Article  MathSciNet  Google Scholar 

  10. Kohlenbach, U.: On the no-counterexample interpretation. J. Symb. Log. 64, 1491–1511 (1999)

    Article  MathSciNet  Google Scholar 

  11. Kohlenbach, U.: Some computational aspects of metric fixed point theory. Nonlinear Anal. 61(5), 823–837 (2005)

    Article  MathSciNet  Google Scholar 

  12. Kohlenbach, U.: Some logical metatheorems with applications in functional analysis. Trans. Amer. Math. Soc. 357, 89–128 (2005)

    Article  MathSciNet  Google Scholar 

  13. Kohlenbach, U.: Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Monographs in Mathematics. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-77533-1

    Book  MATH  Google Scholar 

  14. Kohlenbach, U.: Proof-theoretic methods in nonlinear analysis. In: Proceedings of the ICM 2018, vol. 2, pp. 79–100. World Scientific (2019)

    Google Scholar 

  15. Kohlenbach, U., Koutsoukou-Argyraki, A.: Rates of convergence and metastability for abstract Cauchy problems generated by accretive operators. J. Math. Anal. Appl. 423, 1089–1112 (2015)

    Article  MathSciNet  Google Scholar 

  16. Kohlenbach, U., Leuştean, L.: Effective metastability of Halpern iterates in CAT(0) spaces. Adv. Math. 321, 2526–2556 (2012)

    Article  MathSciNet  Google Scholar 

  17. Kreisel, G.: On the interpretation of non-finitist proofs. Part I. J. Symb. Log. 16, 241–267 (1951)

    MATH  Google Scholar 

  18. Kreisel, G.: On the interpretation of non-finitist proofs, Part II: interpretation of number theory. J. Symb. Log. 17, 43–58 (1952)

    Article  Google Scholar 

  19. Kreisel, G.: Mathematical significance of consistency proofs. J. Symb. Log. 23(2), 155–182 (1958)

    Article  MathSciNet  Google Scholar 

  20. Lombardi, H., Quitté, C.: Commutative Algebra: Constructive Methods: Finite Projective Modules. Springer, Dordrecht (2015). https://doi.org/10.1007/978-94-017-9944-7

    Book  MATH  Google Scholar 

  21. Mulvey, C., Wick-Pelletier, J.: A globalization of the Hahn-Banach theorem. Adv. Math. 89, 1–59 (1991)

    Article  MathSciNet  Google Scholar 

  22. Negri, S., von Plato, J., Coquand, T.: Proof-theoretical analysis of order relations. Arch. Math. Logic 43, 297–309 (2004)

    Article  MathSciNet  Google Scholar 

  23. Oliva, P., Powell, T.: A game-theoretic computational interpretation of proofs in classical analysis. In: Kahle, R., Rathjen, M. (eds.) Gentzen’s Centenary, pp. 501–531. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-10103-3_18

    Chapter  Google Scholar 

  24. Oliva, P., Powell, T.: Spector bar recursion over finite partial functions. Ann. Pure Appl. Log. 168(5), 887–921 (2017)

    Article  Google Scholar 

  25. Persson, H.: An application of the constructive spectrum of a ring. In: Type Theory and the Integrated Logic of Programs. Chalmers University and University of Göteborg (1999). Ph.D. thesis

    Google Scholar 

  26. Powell, T.: On bar recursive interpretations of analysis. Ph.D. thesis, Queen Mary University of London (2013)

    Google Scholar 

  27. Powell, T.: Gödel’s functional interpretation and the concept of learning. In: Proceedings of Logic in Computer Science (LICS 2016), pp. 136–145. ACM (2016)

    Google Scholar 

  28. Powell, T.: Sequential algorithms and the computational content of classical proofs (2018). https://arxiv.org/abs/1812.11003

  29. Richman, F.: Nontrivial uses of trivial rings. Proc. Am. Math. Soc. 103(4), 1012–1014 (1988)

    Article  MathSciNet  Google Scholar 

  30. Rinaldi, D., Schuster, P.: A universal Krull-Lindenbaum theorem. J. Pure Appl. Algebra 220, 3207–3232 (2016)

    Article  MathSciNet  Google Scholar 

  31. Rinaldi, D., Schuster, P., Wessel, D.: Eliminating disjunctions by disjunction elimination. Bull. Symb. Logic 23(2), 181–200 (2017)

    Article  MathSciNet  Google Scholar 

  32. Rinaldi, D., Schuster, P., Wessel, D.: Eliminating disjunctions by disjunction elimination. Indag. Math. (N.S.) 29(1), 226–259 (2018)

    Article  MathSciNet  Google Scholar 

  33. Rinaldi, D., Wessel, D.: Cut elimination for entailment relations. Arch. Math. Log. (2018). https://doi.org/10.1007/s00153-018-0653-0

    Article  MathSciNet  Google Scholar 

  34. Rinaldi, D., Wessel, D.: Extension by conservation. Sikorski’s theorem. Log. Methods Comput. Sci. 14(4:8), 1–17 (2018)

    MathSciNet  MATH  Google Scholar 

  35. Sanders, S.: Metastability and higher-order computability. In: Artemov, S., Nerode, A. (eds.) LFCS 2018. LNCS, vol. 10703, pp. 309–330. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72056-2_19

    Chapter  Google Scholar 

  36. Schlagbauer, K., Schuster, P., Wessel, D.: Der Satz von Hahn-Banach im Rahmen einer allgemeinen Idealtheorie. Confluentes Math, forthcoming

    Google Scholar 

  37. Schuster, P.: Induction in algebra: a first case study. In: 2012 27th Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 581–585. IEEE Computer Society Publications (2012). Proceedings, LICS 2012, Dubrovnik, Croatia

    Google Scholar 

  38. Schuster, P.: Induction in algebra: a first case study. Log. Methods Comput. Sci. 9(3), 20 (2013)

    Article  MathSciNet  Google Scholar 

  39. Schuster, P., Wessel, D.: A general extension theorem for directed-complete partial orders. Rep. Math. Logic 53, 79–96 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Schwichtenberg, H., Seisenberger, M., Wiesnet, F.: Higman’s lemma and its computational content. In: Kahle, R., Strahm, T., Studer, T. (eds.) Advances in Proof Theory. PCSAL, vol. 28, pp. 353–375. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29198-7_11

    Chapter  Google Scholar 

  41. Simpson, S.G.: Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic. Springer, Berlin (1999)

    Book  Google Scholar 

  42. Spector, C.: Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In: Dekker, F.D.E. (ed.) Recursive Function Theory: Proceedings of the Symposia in Pure Mathematics, vol. 5, pp. 1–27. American Mathematical Society, Providence, Rhode Island (1962)

    Google Scholar 

  43. Tao, T.: Soft analysis, hard analysis, and the finite convergence principle. Essay, published as Chap. 1.3 of T. Tao, Structure and Randomness: Pages from Year 1 of a Mathematical Blog, Amer. Math. Soc (2008). Original version available online at http://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/

  44. Wessel, D.: Ordering groups constructively. Commun. Algebra, forthcoming

    Google Scholar 

  45. Yengui, I.: Making the use of maximal ideals constructive. Theor. Comput. Sci. 392, 174–178 (2008)

    Article  MathSciNet  Google Scholar 

  46. Yengui, I.: Constructive Commutative Algebra. LNM, vol. 2138. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-19494-3

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous referees for their detailed comments, which led to a much improved version of the paper.

Author information

Authors and Affiliations

  1. Technische Universität Darmstadt, Darmstadt, Germany

    Thomas Powell

  2. University of Verona, Verona, Italy

    Peter Schuster

  3. University of Trento, Trento, Italy

    Franziskus Wiesnet

Authors
  1. Thomas Powell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Schuster
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Franziskus Wiesnet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Powell .

Editor information

Editors and Affiliations

  1. Department of Philosophy and Religious Studies, Utrecht University, Utrecht, The Netherlands

    Rosalie Iemhoff

  2. Utrecht Institute of Linguistics OTS, Utrecht University, Utrecht, The Netherlands

    Michael Moortgat

  3. Centro de Informática, Universidade Federal de Pernambuco (UFPE), Recife, Brazil

    Ruy de Queiroz

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer-Verlag GmbH Germany, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Powell, T., Schuster, P., Wiesnet, F. (2019). An Algorithmic Approach to the Existence of Ideal Objects in Commutative Algebra. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science(), vol 11541. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59533-6_32

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-59533-6_32

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-59532-9

  • Online ISBN: 978-3-662-59533-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation