A note on generalized Krull domains (Q2874693)

Let \(D\) be an integral domain with quotient field \(K\) and let \(F(D)\) denote the set of nonzero fractional ideals of \(D.\) Set \(I^{-1}=\{x\in K:xI\subseteq D\},~I_{v}=(I^{-1})^{-1}\) and \(I_{t}=\bigcup \{F_{v}:0\neq F\) is a finitely generated subideal of \(I\}.\) The operations \(v\) and \(t\) defining the relations \(A\mapsto A_{v},A_{t}\) are functions on \(F(D)\) that are examples of star operations. A fractional ideal \(I\) is a \(\;v\)-ideal if \( I=I_{v}\) (resp., a \(t\)-ideal if \(I=I_{t})\), \(t\)-invertible if \( (II^{-1})_{t}=D\) and \(v\)-finite if \(I=J_{v}\) for a finitely generated ideal \( J.\) Also an integral ideal \(I\) maximal w.r. t. being a \(t\)-ideal is a prime ideal called a maximal \(t\)-ideal (see sections 32, 34 of \textit{R. Gilmer}'s [Multiplicative ideal theory. York: Marcel Dekker, Inc. (1972; Zbl 0248.13001)] and/or the reviewer's [\textit{M. Zafrullah}, in: Non-Noetherian commutative ring theory. Dordrecht: Kluwer Academic Publishers. 429--457 (2000; Zbl 0988.13003)] for details). One example of the use of star operations in defining new concepts from old is the notion of a Prüfer \(v\)-multiplication domain (PVMD) from Prüfer domains. To see this note that (1) \(D\) is a Prüfer domain (resp., PVMD) if every nonzero ideal is invertible (resp., \(t\)-invertible). (2) \(D\) is a Prüfer domain (resp., PVMD) if and for every maximal ideal (maximal \(t\)-ideal) \(P,\) \(D_{P}\) is a valuation domain. The list can go on. But let's note it wasn't this easy when these results were proved, same with the results mentioned in the paper under review.NEWLINENEWLINEA valuation domain \(V\) is called strongly discrete if for every nonzero prime ideal \(P\) of \(R\) we have \(P^{2}\neq P\) and a Prüfer domain \(D\) is called strongly discrete if for every nonzero prime ideal \(P\) of \(D\) we have \(D_{P}\) strongly discrete. It turns out that a Prüfer domain \(D\) is strongly discrete if and only if for every nonzero prime ideal \(P\) we have \(P^{2}\neq P.\) A Prüfer domain \(D\) is called a generalized Dedekind domain if \(D\) is strongly discrete and every nonzero prime ideal of \(D\) is the radical of a finitely generated ideal. Generalized Dedekind domains were introduced by \textit{N. Popescu} in [Rev. Roum. Math. Pures Appl. 29, 777--786 (1984; Zbl 0564.13011)] and further studied by \textit{M. Fontana} and \textit{N. Popescu} in [Commun. Algebra 23, No. 12, 4521--4533 (1995; Zbl 0835.13004)] and by \textit{B. Olberding} in [J. Lond. Math. Soc., II. Ser. 80, No. 1, 155--170 (2009; Zbl 1198.13006)]. (These domains are not to be confused with G-Dedekind domains \(D\) of the reviewer, characterized by: \(A_{v}\) is invertible for each nonzero ideal \(A\) [Mathematika 33, 285--295 (1986; Zbl 0613.13001)].) The point is, the first author of the paper under review introduced, via the \(t\)-operation, the strongly discrete PVMDs as PVMDs whose prime \(t\)-ideal \(P\) satisfy \((P^{2})_{t}\neq P,\) and generalized Krull domains as strongly discrete PVMDs whose prime \(t\)-ideals are each a radical of a \(v\)-finite ideal [Commun. Algebra 30, No. 8, 3723--3742 (2002; Zbl 1054.13001)].NEWLINENEWLINEIn the paper under review, the authors develop the theory of generalized Krull domains still further. They say that a domain \(D\) has Mori spectrum if the induced topology on \(t-\mathrm{Spec}(D)\) by the Zariski topology on \(\mathrm{Spec}(D)\) is Noetherian. Here \(t-\mathrm{Spec}(D)\) denotes the set of prime \(t\)-ideals of \(D.\) They show via Proposition 2.1 that \(D\) having Mori spectrum is equivalent to every radical \(t\)-ideal being \(v\)-finite. Thus showing in Theorem 2.2 that \(D \) is a generalized Krull domain if and only if \(D\) is a strongly discrete PVMD with Mori spectrum. They also show in Proposition 2.9 that a generalized Krull domain is a strongly discrete PVMD of finite \(t\)-character (every nonzero element belongs to at most a finite number of maximal \(t\)-ideals). They also study generalized Krull domains with torsion class group. Next, in section 4, the authors show that if \(L\) is a finite extension of \(K\), then the integral closure \(T\) of \(D\) in \(L\) is a generalized Krull domain, if \(D\) is. Some characterizations of generalized Krull domains are included at the end. In short the paper is laden with interesting results.NEWLINENEWLINEReviewer's remarks: Recall that a PVMD \(D\) with torsion class group is an almost GCD domain: For each pair \(a,b\in D\backslash \{0\}\) there is a positive integer \(n\) such that \(a^{n}D\cap b^{n}D\) is principal, see e.g. Theorem 3.1 of \textit{D. D. Anderson} and the reviewer in [J. Algebra 142, No. 2, 285--309 (1991; Zbl 0749.13013)]. This information, coupled with a study of \textit{T. Dumitrescu} et al. [J. Algebra 245, No. 1, 161--181 (2001; Zbl 1094.13537)], can, hopefully, give a more satisfying description of the nonzero elements of a generalized Krull domain with torsion class group.