Almost splitting sets in integral domains, II

Let D be an integral domain. A saturated multiplicative subset S of D is an almost splitting set if, for each 0≠d∈D, there exists a positive integer n=n(d) such that dⁿ=st for some s∈S and t∈D which is v-coprime to each element of S. We show that every upper to zero in DX] contains a primary element if and only if D?{0} is an almost splitting set in DX], if and only if D is a UMT-domain and Cl(DX]) is torsion. We also prove that DX] is an almost GCD-domain if and only if D is an almost GCD-domain and Cl(DX]) is torsion. Using this result, we construct an integral domain D such that Cl(D) is torsion, but Cl(DX]) is not torsion.