T-SPLITTING SETS S OF AN INTEGRAL DOMAIN D SUCH THAT DSIS A FACTORIAL DOMAIN

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).