The logical complexity of finitely generated commutative rings

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 $\mathbb N$.

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