\(m\)-canonical ideals in integral domains. II. (Q2741188)

For part I see \textit{W. J. Heinzer}, \textit{J. A. Huckaba} and \textit{I. J. Papick}, Commun. Algebra 26, 3021--3043 (1998; Zbl 0920.13001).NEWLINENEWLINE From the paper: Let \(R\) be an integral domain. A nonzero ideal \(I\) of \(R\) is said to be \(m\)-canonical if \((I:(I:J))=J\) for each nonzero ideal \(J\) of \(R\). When \(R\) is a certain pullback of a valuation domain, we characterize when \(R\) has a finitely generated \(m\)-canonical ideal. We also study domains in which each nondivisorial ideal has the form \(LQ\), where \(Q\) is a product of maximal ideals, and \(L\) is invertible or merely divisorial. For example, let \(R\) be a Prüfer domain with only finitely many idempotent maximal ideals \(M_1\dots,M_t\). We show (theorems 4.5 and 4.6) that the set \(\{LQ\mid L\) is invertible and \(Q\) is a product of \(M_i\}\) is the set of nondivisorial ideals of \(R\) if and only ifNEWLINENEWLINE (1) \(R\) is \(h\)-local,NEWLINENEWLINE (2) if \(t\geq 1\), then each finitely generated maximal ideal of \(R\) has height 1, andNEWLINENEWLINE (3) if \(t\geq 2\), then each divisorial ideal which is contained in exactly one of the \(M_i\) is invertible.NEWLINENEWLINE We also show theorems 4.10 and 4.12 that in a semilocal Prüfer domain \(R\), each nondivisorial ideal has the form \(LQ\), with \(L\) divisorial and \(Q\) a product of maximal ideals, if and only if \(R\) is \(h\)-local.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00058].