# Maximal ideals in the ring of regulous functions are not finitely generated

## TL;DR

This paper investigates the algebraic structure of regulous functions on real affine spaces, proving that maximal ideals are not finitely generated in dimensions two and higher, and extends this to smooth algebraic varieties.

## Contribution

It introduces a regulous Nullstellensatz and demonstrates that maximal ideals in the ring of regulous functions are not finitely generated for dimensions two and above.

## Key findings

- Maximal ideals in the ring of regulous functions on $\,\mathbb{R}^N$ are not finitely generated for $N\geq 2$.
- The paper establishes a regulous Nullstellensatz based on substitution and Artin-Lang properties.
- The non-finite generation result extends to smooth real affine algebraic varieties of dimension at least 2.

## Abstract

The paper consider regulous functions on the real affine space $\mathbb{R}^N$. We shall study some algebraic properties of the ring of those functions. It is presented a proof of the regulous version of Nullstellensatz based on the substitution property and the Artin-Lang property for the considered function ring. We prove that every maximal ideal in the ring of regulous functions on $\mathbb{R}^N$ when $N\geq 2$ is not finitely generated. Finally, we extend the latter result to an arbitrary, smooth, real affine algebraic variety of dimension $d\geq 2$.