Interpretability and uniform definability of integers, and undecidability of reduced indecomposable polynomial rings

Marco Barone; Nicolás Caro; Eudes Naziazeno

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arXiv · 1703.08266

Interpretability and uniform definability of integers, and undecidability of reduced indecomposable polynomial rings

Marco Barone·Nicolás Caro·Eudes Naziazeno

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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; in this direction, we also provide a separate proof of the undecidability of these rings that works uniformly in any characteristic. These definability and undecidability assertions extend a series of results by Raphael Robinson (1951), holding for certain polynomial integral domains, to a more general class. Finally, we show that the rational integers are interpretable in these rings, even in positive characteristic.

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BibTeX RIS

Marco Barone, Nicolás Caro, Eudes Naziazeno. 2020-05-21. Interpretability and uniform definability of integers, and undecidability of reduced indecomposable polynomial rings. https://arxiv.org/abs/1703.08266

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