The Logical Complexity of Finitely Generated Commutative Rings. (27th February 2018) - Aschenbrenner, Matthias; Khélif, Anatole; Naziazeno, Eudes; Scanlon, Thomas
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 $(\mathbb{N}, {+}, {× })$ 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 $\mathbb{N}$ .