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LCM-splitting sets in some ring extensions

Authors:	Tiberiu Dumitrescu Muhammad Zafrullah

Institution:	Facultatea de Matematica, Universitatea Bucuresti, Str. Academiei 14, Bucharest, RO-70190, Romania ; Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085

Abstract:	Let be a saturated multiplicative set of an integral domain . Call 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 is an $R_{2}$ -stable overring of (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 is an lcm splitting set of , then the saturation of in is an lcm splitting set in . Consequently, if is Noetherian and $p\in D$ is a (nonzero) prime element, then is also a prime element of the integral closure of . Also, if is Noetherian, is generated by prime elements of and if the integral closure of $D_{S}$ is a UFD, then so is the integral closure of .

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

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