Korean J. Math. Vol. 21 No. 4 (2013) pp.455-462
DOI: https://doi.org/10.11568/kjm.2013.21.4.455

$t$-splitting sets $S$ of an integral domain $D$ such that $D_S$ is a factorial domain

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Published: Dec 11, 2013
Mathematics Subject Classification:
13A15, 13G05

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Gyu Whan Chang
Department of Mathematics, Incheon National University, Incheon 406-772, Republic of Korea

Abstract

Let $D$ be an integral domain, $S$ be a saturated multiplicative subset of $D$ such that $D_S$ is a factorial domain, $\{X_{\alpha}\}$ be a nonempty set of indeterminates, and $D[\{X_{\alpha}\}]$ be the polynomial ring over $D$. We show that $S$ is a splitting (resp., almost splitting, $t$-splitting) set in $D$ if and only if every nonzero prime $t$-ideal of $D$ disjoint from $S$ is principal (resp., contains a primary element, is $t$-invertible). We use this result to show that $D \setminus \{0\}$ is a splitting (resp., almost splitting, $t$-splitting) set in $D[\{X_{\alpha}\}]$ if and only if $D$ is a GCD-domain (resp., UMT-domain with $Cl(D[\{X_{\alpha}\}])$ torsion, UMT-domain).


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