Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T20:40:44.214Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIFORM DEFINABILITY OF INTEGERS IN REDUCED INDECOMPOSABLE POLYNOMIAL RINGS

Part of: Ring extensions and related topics Model theory Conditions on elements Arithmetic rings and other special rings

Published online by Cambridge University Press:  05 October 2020

MARCO BARONE ,
NICOLÁS CARO  and
EUDES NAZIAZENO
MARCO BARONE
Affiliation:
DEPARTAMENTO DE MATEMÁTICA UNIVERSIDADE FEDERAL DE PERNAMBUCO AVENIDA JORNALISTA ANÍBAL FERNANDES S/N - CIDADE UNIVERSITÁRIA RECIFE/PE 50740-560, BRAZILE-mail: [email protected]: [email protected]: [email protected]
NICOLÁS CARO
Affiliation:
DEPARTAMENTO DE MATEMÁTICA UNIVERSIDADE FEDERAL DE PERNAMBUCO AVENIDA JORNALISTA ANÍBAL FERNANDES S/N - CIDADE UNIVERSITÁRIA RECIFE/PE 50740-560, BRAZILE-mail: [email protected]: [email protected]: [email protected]
EUDES NAZIAZENO
Affiliation:
DEPARTAMENTO DE MATEMÁTICA UNIVERSIDADE FEDERAL DE PERNAMBUCO AVENIDA JORNALISTA ANÍBAL FERNANDES S/N - CIDADE UNIVERSITÁRIA RECIFE/PE 50740-560, BRAZILE-mail: [email protected]: [email protected]: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with signature $(0,1,+,\cdot )$. In the characteristic zero case, the claim implies that the full theory is undecidable, for rings of the referred type. This extends a series of results by Raphael Robinson, holding for certain polynomial integral domains, to a more general class.

Keywords

definabilitypolynomial ringslanguage of rings

MSC classification

Primary: 03C40: Interpolation, preservation, definability
Secondary: 13B25: Polynomials over commutative rings 13F99: None of the above, but in this section 16U99: None of the above, but in this section
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1376 - 1402
Check for updates
Copyright
© The Association for Symbolic Logic 2020

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

Arthan, R., Is $\mathbb{Z}$ first-order definable in (the ring) $\mathbb{Z}\times \mathbb{Z}$?. Mathematics Stack Exchange, version: 2016-01-23. Available at https://math.stackexchange.com/q/1623441.Google Scholar
Aschenbrenner, M., Khélif, A., Naziazeno, E., and Scanlon, T., The logical complexity of finitely generated commutative rings. International Mathematics Research Notices, vol. 2020 (2018), no. 1, pp. 112–166.Google Scholar
Blass, A., Injectivity, projectivity, and the axiom of choice. Transactions of the American Mathematical Society, vol. 255 (1979), pp. 31–59.CrossRefGoogle Scholar
Cohn, P. M., Basic Algebra: Groups, Rings and Fields, Springer-Verlag London, Ltd., London, 2003.CrossRefGoogle Scholar
Denef, J., The Diophantine problem for polynomial rings and fields of rational functions. Transactions of the American Mathematical Society, vol. 242 (1978), pp. 391–399.CrossRefGoogle Scholar
Denef, J., The Diophantine problem for polynomial rings of positive characteristic, Logic Colloquium ’78 (Mons, 1978) (Boffa, M., Dalen, D., and Mcaloon, K., editors), Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, Amsterdam, 1979, pp. 131–145.Google Scholar
Eisenbud, D., Commutative Algebra: With a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
Good, C. and Tree, I. J., Continuing horrors of topology without choice. Topology and Its Applications, vol. 63 (1995), no. 1, pp. 79–90.CrossRefGoogle Scholar
Hodges, W., Six impossible rings. Journal of Algebra, vol. 31 (1974), pp. 218–244.CrossRefGoogle Scholar
Hodges, W., Krull implies Zorn. The Journal of the London Mathematical Society. Second Series, vol. 19 (1979), no. 2, pp. 285–287.Google Scholar
Hodges, W., Model theory, Encyclopedia of Mathematics and Its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
Howard, P. and Rubin, J. E., Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, RI, 1998.CrossRefGoogle Scholar
Hungerford, T. W., Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, New York, 1980, reprint of the 1974 original.CrossRefGoogle Scholar
Kassel, C., Quantum Groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995.Google Scholar
Koenigsmann, J., Undecidability in number theory, Model Theory in Algebra, Analysis and Arithmetic (Macpherson, H. D. and Toffalori, C., editors), Lecture Notes in Mathematics, vol. 2111, Springer, Heidelberg, 2014, pp. 159–195.Google Scholar
Matijasevič, J. V., The Diophantineness of enumerable sets. Doklady Akademii Nauk SSSR, vol. 191 (1970), pp. 279–282.Google Scholar
Nies, A., Describing groups. The Bulletin of Symbolic Logic, vol. 13 (2007), no. 3, pp. 305–339.CrossRefGoogle Scholar
Pheidas, T., Decision problems in algebra and analogues of Hilbert’s tenth problem, Model Theory with Applications to Algebra and Analysis, vol. 2, (Chatzidakis, Z., Macpherson, D., Pillay, A., and Wilkie, A., editors), London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 207–235.CrossRefGoogle Scholar
Pheidas, T. and Zahidi, K., Undecidable existential theories of polynomial rings and function fields. Communications in Algebra, vol. 27 (1999), no. 10, pp. 4993–5010.CrossRefGoogle Scholar
Poonen, B., Undecidability in number theory. Notices of the American Mathematical Society, vol. 55 (2008), no. 3, pp. 344–350.Google Scholar
Robinson, J., Definability and decision problems in arithmetic, this Journal, vol. 14 (1949), pp. 98–114.Google Scholar
Robinson, R. M., Undecidable rings. Transactions of the American Mathematical Society, vol. 70 (1951), pp. 137–159.CrossRefGoogle Scholar
Sawin, W., Rings for which no polynomial induces the zero function. MathOverflow, version: 2014-03-21. Available at https://mathoverflow.net/q/161057 .Google Scholar
Shlapentokh, A., Diophantine definitions for some polynomial rings. Communications on Pure and Applied Mathematics, vol. 43 (1990), no. 8, pp. 1055–1066.CrossRefGoogle Scholar
Shlapentokh, A., Defining integers. The Bulletin of Symbolic Logic, vol. 17 (2011), no. 2, pp. 230–251.CrossRefGoogle Scholar
Utreras, J., Interpreting arithmetic in the first-order theory of addition and coprimality of polynomial rings, this Journal, vol. 84 (2019), no. 3, pp. 1194–1214.Google Scholar
van Dobben de Bruyn, R., Do there exist nonzero identically vanishing polynomials over infinite (or characteristic zero) reduced indecomposable commutative rings? MathOverflow, version: 2017-05-02. Available at https://mathoverflow.net/q/268753.Google Scholar