The Logical Complexity of Finitely Generated Commutative Rings. (27th February 2018)

• Record Type: Journal Article
• Authors: Aschenbrenner, Matthias; Khélif, Anatole; Naziazeno, Eudes; Scanlon, Thomas
• Is Part Of: International mathematics research notices. Volume 2020:Number 1(2020)
• Page Start: 112
• Page End: 166
• Publication Date: 2018-02-27
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rny023
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: We characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with $(\mathbb{N}, {+}, {\times })$ if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in $\mathbb{Z}$ . Notably, by constructing a nontrivial derivation on a nonstandard model of arithmetic we show that the ring of dual numbers over $\mathbb{Z}$ is not bi-interpretable with $\mathbb{N}$ .