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UNIFORM DEFINABILITY OF INTEGERS IN REDUCED INDECOMPOSABLE POLYNOMIAL RINGS

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

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, BRAZIL E-mail: marco.barone@ufpe.br E-mail: jorge.caro@ufpe.br E-mail: eudes.naziazeno@ufpe.br
NICOLÁS CARO
Affiliation:
DEPARTAMENTO DE MATEMÁTICA UNIVERSIDADE FEDERAL DE PERNAMBUCO AVENIDA JORNALISTA ANÍBAL FERNANDES S/N - CIDADE UNIVERSITÁRIA RECIFE/PE 50740-560, BRAZIL E-mail: marco.barone@ufpe.br E-mail: jorge.caro@ufpe.br E-mail: eudes.naziazeno@ufpe.br
EUDES NAZIAZENO
Affiliation:
DEPARTAMENTO DE MATEMÁTICA UNIVERSIDADE FEDERAL DE PERNAMBUCO AVENIDA JORNALISTA ANÍBAL FERNANDES S/N - CIDADE UNIVERSITÁRIA RECIFE/PE 50740-560, BRAZIL E-mail: marco.barone@ufpe.br E-mail: jorge.caro@ufpe.br E-mail: eudes.naziazeno@ufpe.br
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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

Information

Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 4 , December 2020 , pp. 1376 - 1402
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Copyright
© The Association for Symbolic Logic 2020

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References

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