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UNIFORM DEFINABILITY OF INTEGERS IN REDUCED INDECOMPOSABLE POLYNOMIAL RINGS

Published online by Cambridge University Press:  05 October 2020

MARCO BARONE
Affiliation:
DEPARTAMENTO DE MATEMÁTICA UNIVERSIDADE FEDERAL DE PERNAMBUCO AVENIDA JORNALISTA ANÍBAL FERNANDES S/N - CIDADE UNIVERSITÁRIA RECIFE/PE 50740-560, BRAZILE-mail: marco.barone@ufpe.brE-mail: jorge.caro@ufpe.brE-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, BRAZILE-mail: marco.barone@ufpe.brE-mail: jorge.caro@ufpe.brE-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, BRAZILE-mail: marco.barone@ufpe.brE-mail: jorge.caro@ufpe.brE-mail: eudes.naziazeno@ufpe.br

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.

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Articles
Copyright
© The Association for Symbolic Logic 2020

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