LCM-splitting sets in some ring extensions.
In: Proceedings of the American Mathematical Society, Jg. 130 (2002-06-01), Heft 6, S. 1639-1644
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Zugriff:
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 $D$. [ABSTRACT FROM AUTHOR]
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LCM-splitting sets in some ring extensions.
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Autor/in / Beteiligte Person: | Dumitrescu, Tiberiu ; Zafrullah, Muhammad |
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Zeitschrift: | Proceedings of the American Mathematical Society, Jg. 130 (2002-06-01), Heft 6, S. 1639-1644 |
Veröffentlichung: | 2002 |
Medientyp: | academicJournal |
ISSN: | 0002-9939 (print) |
DOI: | 10.1090/S0002-9939-01-06301-8 |
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