Splitting sets in integral domains

Let $D$ be an integral domain. A saturated multiplicatively closed subset $S$of $D$ is a splitting set if each nonzero $d\in D$ may be written as $d=sa$ where $s\in S$ and $s'D\cap aD=s'aD$ for all $s'\in S$. We show that if $S$ is a splitting set in $D$, then $SU(D_{N})$ is a splitting set in $D_{N}$, $N$ a multiplicatively closed subset of $D$, and that $S\subseteq D$ is a splitting set in $D[X]\iff S$ is an lcm splitting set of $D$, i.e., $S$ is a splitting set of $D$ with the further property that $sD\cap dD$ is principal for all $s\in S$ and $d\in D$. Several new characterizations and applications of splitting sets are given.