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LCM-splitting sets in some ring extensions
Authors:Tiberiu Dumitrescu   Muhammad Zafrullah
Affiliation:Facultatea de Matematica, Universitatea Bucuresti, Str. Academiei 14, Bucharest, RO-70190, Romania ; Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085
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 $dDcap sD$ are principal ideals for every $din D$ and $sin S$. We show that if $R$ is an $R_{2}$-stable overring of $D$ (that is, if whenever $a,bin D$ and $aDcap bD$ is principal, it follows that $(aDcap bD)R=aRcap bR)$ 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 $pin D$ 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 $D$.

Keywords:lcm-splitting set   $R_{2}$-stable overring   Noetherian domain
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