LCM-splitting sets in some ring extensions

Abstract:

Let $S$ be a saturated multiplicative set of an integral domain $D$. Call $S$ an lcm splitting set if $dD_{S}\cap D$ and $dD\cap sD$ are principal ideals for every $d\in D$ and $s\in S$. We show that if $R$ is an $R_{2}$-stable overring of $D$ (that is, if whenever $a,b\in D$ and $aD\cap bD$ is principal, it follows that $(aD\cap bD)R=aR\cap bR)$ and if $S$ is an lcm splitting set of $D$, then the saturation of $S$ in $R$ is an lcm splitting set in $R$. Consequently, if $D$ is Noetherian and $p\in D$ is a (nonzero) prime element, then $p$ is also a prime element of the integral closure of $D$. Also, if $D$ is Noetherian, $S$ is generated by prime elements of $D$ and if the integral closure of $D_{S}$ is a UFD, then so is the integral closure of $D$.