A Necessary and Sufficient Condition for Uniqueness of Euclidean Division

TL;DR

This paper proves that the classical characterization of Euclidean domains with unique division holds true under the modern definition, confirming the longstanding open question.

Contribution

It establishes that the known necessary and sufficient condition for unique Euclidean division applies to the modern definition of Euclidean domains.

Findings

The classical characterization remains valid under the modern definition.

The result confirms the equivalence of definitions regarding Euclidean domains.

Addresses an open question from the 1960s about division uniqueness.

Abstract

A well-known result from the 1960s characterizes all Euclidean domains in which division is guaranteed to produce a unique quotient and remainder. As this relies on the historical (and more restrictive) definition of a Euclidean domain, the question of whether the result still holds under the modern definition was left open. In this paper, we prove the answer is afirmative.