Prüfer \(v\)-multiplication domains and valuation ideals (Q2839874)

Let \(D\) be an integral domain with quotient field \(K\) and let \(F(D)\) denote the set of nonzero fractional ideals of \(D.\) Call a ring \(R\) an overring of \(D\) if \(D\subseteq R\subseteq K.\) On \(F(D),\) define functions \( A\mapsto \) \(A_{v}=(A^{-1})^{-1}\), \(A\mapsto A_{t}=\bigcup \{F_{v}\): \(F\) nonzero finitely generated (f. g.) subideal of \(A\}\) and \(A_{w}=\{x\in K:xF\subseteq A\) for some f. g. \(F\) with \(F^{-1}=D\}.\) The functions \(v,t\) and \(w\) are examples of star operations which are being used as tools for generalizing various concepts these days. A generalization of a Prüfer domain (every f. g. nonzero ideal is invertible) is the so called Prüfer \(v\)-multiplication domain (P\(v\)MD) (every nonzero f. g. ideal \(A\) is \(t\)-invertible, i.e., \( (AA^{-1})_{t}=D).\) Experience has it that the theory of P\(v\)MDs runs parallel to that of Prüfer domains, more or less. So it is natural to look for a P\(v\)MD analogue of a result on Prüfer domains. See \textit{D. E. Dobbs} et al., [Commun. Algebra 17, No. 11, 2835--2852 (1989; Zbl 0691.13015)] for P\( v\)MD analogues of Prüfer domain results.NEWLINENEWLINEIn the paper under review, the author aims at proving P\(v\)MD analogues of the following results: (a) An integral domain \(D\) is a Prüfer domain if and only if every ideal of \(D\) is an intersection of valuation ideals [\textit{R. Gilmer} and \textit{J. Ohm}, Trans. Am. Math. Soc. 117, 237--250 (1965; Zbl 0133.29203)]). (b) \(D\) is Prüfer if and only if \(D\) is integrally closed and there is an integer \( n>1\) such that \((a,b)^{n}=(a^{n},b^{n}),\) for all \(a,b\in D\) ((4) of Theorem 24.3 of \textit{R. Gilmer}'s [Multiplicative ideal theory. Pure and Applied Mathematics. Vol. 12. New York: Marcel Dekker, Inc. X (1972; Zbl 0248.13001); Corrected reprint Queen's Papers in Pure and Applied Mathematics, 90. Queen's University, Kingston, ON (1992)]. Here an ideal \(I\) is a valuation ideal if \( I=IV\cap D\) for some valuation overring \(V\) of \(D.\)NEWLINENEWLINETo accomplish this, the author introduces the notion of a \(t\)-valuation ideal (an ideal \(I\) such that \(I=IV\cap D\) for some valuation overring \(V\) of \(D\) such that \(V\) is \(t\)-linked over \(D).\) Here an overring \( R\) of \(D\) is \(t\) -linked over \(D\) if for every f. g. ideal \(I\) of \(D\) with \( I^{-1}=D\) we have \((IR)^{-1}=R.\) The author ends up proving, among other things, that (a) \(D\) is a P\(v\)MD if and only if for every ideal \(I\) of \(D,\) \( I_{w}\) is an intersection of \(t\)-valuation ideals of \(D\) and (b) \(D\) is a P\( v \)MD if and only if \(D\) is integrally closed and there is an integer \(n>1\) such that \(((a,b)^{n})_{w}=(a^{n},b^{n})_{w}\) for all \(a,b\in D.\)