PreprintCC BY-SA 4.0arXiv
LCM Splittings in GCD Domains
Abstract
In a GCD ring R, every pair of elements a,b∈ R admits a greatest common divisor gcd(a,b) and a least common multiple 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∈ R for a,b satisfying cd∼lcm(a,b), c| a, d| b, gcd(c,d)∼1. 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.
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BibTeX
@misc{preu2026splittings,
title = {{LCM Splittings in GCD Domains}},
author = {Thomas Preu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23799},
eprint = {2610.23799},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23799},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - LCM Splittings in GCD Domains
AU - Thomas Preu
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23799
UR - https://arxiv.org/abs/2610.23799
AB - In a GCD ring R, every pair of elements a,b∈ R admits a greatest common divisor gcd(a,b) and a least common multiple 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∈ R for a,b satisfying cd∼lcm(a,b), c| a, d| b, gcd(c,d)∼1. 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.
ER -