Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-25T04:13:28.475Z Has data issue: false hasContentIssue false
  • English
  • Français

BOUNDS FOR INDEXES OF NILPOTENCY IN COMMUTATIVE RING THEORY: A PROOF MINING APPROACH

Part of: Proof theory and constructive mathematics General commutative ring theory Ring extensions and related topics

Published online by Cambridge University Press:  05 January 2021

FERNANDO FERREIRA
FERNANDO FERREIRA*
Affiliation:
DEPARTAMENTO DE MATEMÁTICA FACULDADE DE CIÊNCIAS DA UNIVERSIDADE DE LISBOA CAMPO GRANDE, ED. C6, 1749-016LISBOA, PORTUGAL E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well-known that an element of a commutative ring with identity is nilpotent if, and only if , it lies in every prime ideal of the ring. A modification of this fact is amenable to a very simple proof mining analysis. We formulate a quantitative version of this modification and obtain an explicit bound. We present an application. This proof mining analysis is the leitmotif for some comments and observations on the methodology of computational extraction. In particular, we emphasize that the formulation of quantitative versions of ordinary mathematical theorems is of independent interest from proof mining metatheorems.

Keywords

Proof miningquantitative versionsnilpotencycommutative ringsbounded functional interpretation

MSC classification

Primary: 03F10: Functionals in proof theory 13A99: None of the above, but in this section
Secondary: 13B25: Polynomials over commutative rings 03F35: Second- and higher-order arithmetic and fragments
Type
Communication
Information
Bulletin of Symbolic Logic , Volume 26 , Issue 3-4 , December 2020 , pp. 257 - 267
Check for updates
Copyright
© The Association for Symbolic Logic 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Atiyah, M. and MacDonald, I. G., Introduction to Commutative Algebra, Addison-Wesley-Longman, Boston, MA, 1969.Google Scholar
Coste, M., Lombardi, H., and Roy, M.-F., Dynamical method in algebra: effective Nullstellensätze. Annals of Pure and Applied Logic, vol. 111 (2001), no. 3, pp. 203–256.CrossRefGoogle Scholar
Engrácia, P., Proof-theoretic studies on the bounded functional interpretation, PhD thesis, Universidade de Lisboa, 2009.Google Scholar
Engrácia, P. and Ferreira, F., Bounded functional interpretation with an abstract type, Contemporary Logic and Computing (Rezuş, A., editor), vol. 1 of Landscapes in Logic, College Publications, London, 2020, pp. 87–112.Google Scholar
Ferreira, F., A most artistic package of a jumble of ideas. Dialectica, vol. 62 (2008), pp. 205–222. Special Issue: Gödel’s Dialectica interpretation. Guest editor: Thomas Strahm.CrossRefGoogle Scholar
Ferreira, F., Injecting uniformities into Peano arithmetic. Annals of Pure and Applied Logic, vol. 157 (2009), pp. 122–129.CrossRefGoogle Scholar
Ferreira, F., Leustean, L., and Pinto, P., On the removal of weak compactness arguments in proof mining. Advances in Mathematics, vol. 354 (2019), Article 106728.CrossRefGoogle Scholar
Ferreira, F. and Oliva, P., Bounded functional interpretation. Annals of Pure and Applied Logic, vol. 135 (2005), pp. 73–112.CrossRefGoogle Scholar
Friedman, H., Simpson, S., and Smith, R.. Countable algebra and set existence axioms. Annals of Pure and Applied Logic, 25(2):141–181, 1983.CrossRefGoogle Scholar
Harrison-Trainor, M., Klys, J., and Moosa, R., Nonstandard methods for bounds in differential polynomial rings. Journal of Algebra, vol. 360 (2012), pp. 71–86.CrossRefGoogle Scholar
Kohlenbach, U., Analysing proofs in analysis, Logic: from Foundations to Applications (Hodges, W., Hyland, M., Steinhorn, C., and Truss, J., editors), Oxford University Press, Oxford, 1996, pp. 225–260.Google Scholar
Kohlenbach, U., Some logical metatheorems with applications in functional analysis. Transactions of the American Mathematical Society, vol. 357 (2005), pp. 89–128.CrossRefGoogle Scholar
Kohlenbach, U., Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2008.Google Scholar
Kohlenbach, U., On quantitative versions of theorems due to F. E. Browder and R. Wittmann. Advances in Mathematics, vol. 226 (2011), pp. 2764–2795.CrossRefGoogle Scholar
Kohlenbach, U., Proof-theoretic methods in nonlinear analysis, Proceedings of the International Congress of Mathematicians (Sirakov, B., de Sousa, P. Ney, and Viana, M., editors, ICM 2018). World Scientific, Singapore, 2019.Google Scholar
Powell, T., Schuster, P., and Wiesnet, F., An algorithmic approach to the existence of ideal objects in commutative algebra, Logic, Language, Information, and Computation (Iemhoff, R., Moortgat, M., and de Queiroz, R., editors), Springer, Berlin and Heidelberg, 2019, pp. 533–549.CrossRefGoogle Scholar
Richman, F., Nontrivial uses of trivial rings. Proceedings of the American Mathematical Society, vol. 103 (1988), no. 4, pp. 1012–1014.CrossRefGoogle Scholar
Simmons, W. and Towsner, H., Proof mining and effective bounds in differential polynomial rings. Advances in Mathematics, vol. 343 (2019), pp. 567–623.CrossRefGoogle Scholar
Simpson, S. G., Subsystems of Second Order Arithmetic. Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999.CrossRefGoogle Scholar
van den Dries, L. and Schmidt, K., Bounds in the theory of polynomial rings over fields. A nonstandard approach. Inventiones Mathematicae, vol. 76 (1984), pp. 77–91.CrossRefGoogle Scholar