The ring D+XDs[X] and t-splitting sets

Let D be an integral domain, S a multiplicatively closed subset of D, and * a finite character star-operation on D. We say that S is a *-splitting set if for each 0 not equal d is an element of D, there exist integral ideals A and B of D with (d) = (AB)*, where A* boolean AND sD = sA* for all s is a n element of S and B* boolean AND S # (O) over bar. We show that D-(S) = D + XDS[X] is a PVMD (resp., GGCD domain) if and only if D is a PVMD (resp., GGCD domain) and S is a t-splitting (resp., d-splitting) subset of D. Let S be a t-splitting set of D and let T = {A(1)...An\ each A(i) = d(i)D(S) boo lean AND D for some nonzero d(i) is an element of D}. Then D = D-S boolean AND D-T. We relate the t-operation on D to the t-operation on D-S and D-T.