Semistar ascending chain conditions over power series rings (Q2054787)

Let \(D\) be an integral domain, \(\star\) a semistar operation on \(D\) and \(\star_f\) the semistar operation of finite type associated to \(\star.\) \textit{S. El Baghdadi} et al. [J. Pure Appl. Algebra 193, No. 1--3, 27--60 (2004; Zbl 1081.13003)] introduced the notion of semistar Noetherian domains. In the second section of this paper, the authors associate to a semistar operation \(\star\) on \(D\), a semistar operation \(\ast\) on \(D[[X]]\). They show that if \(D\) is a semistar Noetherian domain and if \(D\) satisfies the \(\star\)-finite property, that is each nonzero element \(x\) of \(D\) belongs to only finitely many quasi-\(\star\)-maximal ideal of \(D\), then \(D[[X]]\) is a \(\ast\)-Noetherian domain. In the third section, the authors are interested in domains satisfying the ascending chain condition on radical quasi semistar ideals. This notion is defined by \textit{P. Sahandi} [Rocky Mt. J. Math. 40, No. 3, 1039--1049 (2010; Zbl 1200.13008)]. They give sufficient conditions on \(D\) and the semistar operation on \(D\) for \(D[[X]]\) to satisfy the ascending chain condition on radical quasi-\(\ast\)-ideals.