Other Examples of Principal Ideal Domains that are not Euclidean Domains

TL;DR

The paper tackles the problem of finding principal ideal domains that are not Euclidean domains by constructing an explicit ring $A = F[X,Y]/(X^2+Y^2+1)$ over a real closed field $F$ and proving it is a PID while failing to be Euclidean. It uses unit analysis and a detailed prime-ideal classification to show every nonzero prime ideal is generated by a linear form and that all quotients $A/\overline{J}$ are isomorphic to $F(\sqrt{-1})$, making primes maximal. A self-contained argument based on a Euclidean-domain criterion shows that such a ring cannot be Euclidean, since it would force $-1$ to be a square in $F$. By varying the real closed field $F$, the construction yields a family of non-isomorphic PIDs that are not Euclidean domains, extending the landscape of explicit counterexamples beyond classical cases.

Abstract

It is a well-known and easily established fact that every Euclidean domain is also a principal ideal domain. However, the converse statement is not true, and this is usually shown by exhibiting as a counterexample the ring of algebraic integers in a certain, very specific quadratic field, and the proof that this works is quite unnatural and technical. In this article, we will present a family of counterexamples constructed using real closed fields.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4
• Definition 5
• Theorem 6
• Theorem 7
• Proposition 9
• proof
• Theorem 10