On \(S\)-strong Mori domains (Q404247)

Throughout this review \(D\) is an integral domain and \(S\) is a (not necessarily saturated) multiplicative subset of \(D\).NEWLINENEWLINE\textit{D. D. Anderson} and \textit{T. Dumitrescu} [Commun. Algebra 30, No. 9, 4407--4416 (2002; Zbl 1060.13007)] introduced the concepts of \(S\)-Noetherian rings and \(S\)-principal ideal rings. A commutative ring \(R\) is called \(S\)-Noetherian ring (\(S\)-principal ideal ring) if each ideal of \(R\) is \(S\)-finite (respectively \(S\)-principal), i.~e., for each ideal \(I\) of \(R\) , there exist an \(s\in S\) and a finitely generated (respectively principal) ideal \(J\) of \(R\) such that \(sI\subset J\subset I\).NEWLINENEWLINEThe main purpose of the paper under review is to introduce and to investigate the notions of \(S\)-strong Mori domains, \(S\)-strong Mori modules and \(S\)-factorial domains. An integral domain \(D\) is called an \(S\)-strong Mori domain (\(S\)-factorial domain) if each ideal of \(D\) is \(S\)-\(w\)-finite (respectively, \(S\)-\(w\)-principal). A fractional ideal \(I\) of \(D\) is called of \(w\)-finite type if \(I=J_w\) for some finitely generated ideal \(J\) of \(D\) (\(J_w=\{x\in K|xF\subseteqq J\}\) for some finitely generated ideal \(F\) of \(D\) such that \((F^{-1})^{-1}=D\) and \(K\) is the quotient field of \(D\)).NEWLINENEWLINEIn Section 1 the authors study basic properties of \(S\)-strong Mori domains and \(S\)-factorial domains.NEWLINENEWLINEIn Section 2 they prove that if \(S\) is an anti-Archimedean subset of \(D\), then \(D\) is an \(S\)-strong Mori if and only if the polynomial ring \(D[X]\) is an \(S\)-strong Mori domain, if and only if the \(t\)-Nagata ring \(D[X]_{N_v}\) is an \(S\)-strong Mori domain, if and only if \(D[X]_{N_v}\) is an \(S\)-Noetherian domain.NEWLINENEWLINEThe Section~3 is devoted to study the Cohen type theorem for \(S\)-factorial domains and to give the new characterizations of Krull domains.