Let D be an integral domain, S be a saturated multi-plicative 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 $D{ackslash}{0}$ 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).