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

## TL;DR

This paper establishes the first-order definability of prime subrings in certain polynomial rings with reduced, indecomposable coefficient rings, and proves their undecidability and interpretability of integers, extending classical results to broader classes.

## Contribution

It introduces a uniform first-order definability of prime subrings in polynomial rings with reduced, indecomposable coefficients and proves their undecidability and integer interpretability.

## Key findings

- Prime subring definability in polynomial rings
- Undecidability of the theories of these rings
- Interpretability of integers in the rings

## 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.