Almost splitting sets in integral domains

Let A be an integral domain, S a saturated multiplicative subset of A, and N(S)={0≠x∈A|(x,s)v=A for all s∈S}. Then S is called an almost splitting set if for each 0≠d∈A, there is an integer n=n(d)⩾1 such that dn=st for some s∈S and t∈N(S). Let B be an overring of A, X an indeterminate over B, R=A+XB[X], and D=A+X2B[X]. In this paper, we study almost splitting sets and show that D is an AGCD-domain if and only if R is an AGCD-domain and char(A)≠0. As a corollary, we have that D is an AGCD-domain if A is an integrally closed AGCD-domain, char(A)≠0, and B=AS, where S is an almost splitting set of A.