LCM Splittings in GCD Domains

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.