Ascending nets of Prufer v-multiplication domains

Let D be an integral domain, S C D a multiplicative set such that aD(S) boolean AND D is a principal ideal for each a is an element of D and let D-(S) = boolean ORsis an element ofSD[X/s]. It is known that if D is a Prufer v-multiplication domain (resp., generalized GCD domain, GCD domain), then so is D-(S) respectively. When D is a Noetherian domain, we obtain a similar result for the power series analog D-((S)) = boolean ORsis an element ofSD[[X/s]] of D-(S). Our approach takes care simultaneously of both cases D-(S) and D-((S)).