Quasi-Euclidean subrings of Q[x]

TL;DR

This paper constructs specific subrings of Q[x] using nonstandard analysis, demonstrating the existence of quasi-Euclidean subrings that are not k-stage Euclidean for any norm, with diverse algebraic properties.

Contribution

It introduces a novel nonstandard model approach to identify quasi-Euclidean subrings of Q[x] that are not k-stage Euclidean, expanding understanding of Euclidean-like domains.

Findings

Existence of quasi-Euclidean subrings not k-stage Euclidean for any norm

Construction of such subrings can yield PID or non-UFD structures

There are 2^{ ext{omega}} non-isomorphic such domains

Abstract

Using a nonstandard model of Peano arithmetic, we show that there are quasi-Euclidean subrings of Q[x] which are not k-stage Euclidean for any norm and positive integer k. These subrings can be either PID or non-UFD, depending on the choice of parameters in our construction. In both cases, there are 2^{\omega} such domains up to ring isomorphism.