Almost splitting sets in integral domains, II

Let DD be an integral domain. A saturated multiplicative subset SS of DD is an almost splitting set   if, for each 0≠d∈D0≠d∈D, there exists a positive integer n=n(d)n=n(d) such that dn=stdn=st for some s∈Ss∈S and t∈Dt∈D which is vv-coprime to each element of SS. We show that every upper to zero in D[X]D[X] contains a primary element if and only if D∖{0}D∖{0} is an almost splitting set in D[X]D[X], if and only if DD is a UMT-domain and Cl(D[X])Cl(D[X]) is torsion. We also prove that D[X]D[X] is an almost GCD-domain if and only if DD is an almost GCD-domain and Cl(D[X])Cl(D[X]) is torsion. Using this result, we construct an integral domain DD such that Cl(D)Cl(D) is torsion, but Cl(D[X])Cl(D[X]) is not torsion.