t-SPLITTING SETS S OF AN INTEGRAL DOMAIN D SUCH THAT DS IS A FACTORIAL DOMAIN
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Let D be an integral domain, S be a saturated multi-plicative subset of D such that DS is a factorial domain, ｛Xα｝ bea nonempty set of indeterminates, and D[｛Xα｝] be the polynomialring 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 Ddisjoint from S is principal (resp., contains a primary element, ist-invertible). We use this result to show that D＼｛0｝is a splitting(resp., almost splitting, t-splitting) set in D[｛Xα｝] if and only if Dis a GCD-domain (resp., UMT-domain with Cl(D[｛Xα｝]) torsion,UMT-domain).
Abstract
Let D be an integral domain, S be a saturated multi-plicative subset of D such that DS is a factorial domain, ｛Xα｝ bea nonempty set of indeterminates, and D[｛Xα｝] be the polynomialring 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 Ddisjoint from S is principal (resp., contains a primary element, ist-invertible). We use this result to show that D＼｛0｝is a splitting(resp., almost splitting, t-splitting) set in D[｛Xα｝] if and only if Dis a GCD-domain (resp., UMT-domain with Cl(D[｛Xα｝]) torsion,UMT-domain).
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