Year: 2026
Type: preprint
Abstract: In a GCD ring $R$, every pair of elements $a,b\in R$ admits a greatest common divisor $\text{gcd}(a,b)$ and a least common multiple $\text{lcm}(a,b)$. Motivated by a question on Mathematics Stack Exchange by Ash Seifert~\cite{Sei22}, I introduce the notion of an lcm splitting $c,d\in R$ for $a,b$ satisfying $cd\sim\text{lcm}(a,b), c\mid a, d\mid b, \text{gcd}(c,d)\sim1$. I show that adequate Bézout domains always admit lcm splittings, give examples of Bézout domains in which some pair has no lcm splitting, and make further remarks on this seemingly new property.
Source: arXiv (Cornell University)
Author: Preu, Thomas
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222936316