The logical complexity of finitely generated commutative rings

TL;DR

This paper characterizes finitely generated commutative rings that are bi-interpretable with arithmetic, linking their algebraic structure to topological properties of their prime ideals and nilradical.

Contribution

It provides a precise characterization of when such rings are bi-interpretable with natural numbers, including a construction showing the ring of dual numbers is not bi-interpretable.

Findings

Bi-interpretability depends on the nonempty, connected space of non-maximal prime ideals.

The nilradical's nontrivial annihilator in a7Z is crucial for bi-interpretability.

The ring of dual numbers over a7Z is not bi-interpretable with a7N due to a nontrivial derivation.

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 $\mathbb N$.