The A+XB[X] construction from Prüfer v-multiplication domains

Let A⊆B be an extension of integral domains, X be an indeterminate over B, and R=A+XB[X]. We prove that if B is t-flat over A, then R is a PvMD if and only if A is a PvMD and B=AS for S a t-splitting set of ideals of A. We also prove that R is a GGCD domain if and only if A is a GGCD domain and B=AS for S a d-splitting set of ideals of A. Finally, we use this result to recover that R is a GCD domain if and only if A is a GCD domain and B=AS for some splitting set S of A.