Semistar ascending chain conditions over power series rings

Abstract

Let D be an integral domain. We associate to a semistar operation $$\star $$ on D, a semistar operation $$*$$ on D[[X]]. We prove that if D satisfies the $$\star _f$$-finite character property then D is $$\widetilde{\star }$$-Noetherian if and only if D[[X]] is $$*$$-Noetherian. Moreover we give sufficient conditions on D for D[[X]] to satisfy the ascending chain condition on radical quasi-$$*$$-ideals.