Almost Splitting Sets and Agcd Domains

Let D be an integral domain. A multiplicative set S of D is an almost splitting set if for each 0 6= d ∈ D, there exists an n = n(d) with dn = st where s ∈ S and t is v-coprime to each element of S. An integral domain D is an almost GCD (AGCD) domain if for every x, y ∈ D, there exists a positive integer n = n(x, y) such that xnD ∩ ynD is a principal ideal. We prove that the polynomial ring D[X ] is an AGCD domain if and only if D is an AGCD domain and D[X ] ⊆ D0[X] is a root extension, where D0 is the integral closure of D. We also show that D + XDS [X ] is an AGCD domain if and only if D and DS [X] are AGCD domains and S is an almost splitting set.