The special trace property (Q2785925)

Let \(G=\prod_{n=1}^\infty G_n\), where each \(G_n\) is a subgroup of the reals and let \(V\) be a valuation domain with the corresponding value group \(G\). Then the maximal ideal \(M\) of \(V\) is unbranched but for each non-invertible ideal \(I\) of \(V\), \(I=rQ\), for some \(r\in V\), and some primary ideal \(Q\) with \(\sqrt Q=II^{-1}\). This result gives a negative answer to a question posed by \textit{D. D. Anderson}. The reasoning is done through translation of statements about ideals in a valuation ring to the corresponding statements in the value group. NEWLINENEWLINENEWLINEThe second part of the paper deals with special properties and special rings. The authors have shown that a Mori or a Noetherian domain has the special trace property iff it has the trace property. Let \(R\) be a Prüfer TP domain; then NEWLINENEWLINENEWLINE(a) if dim \(R=1\), then \(R\) is an STP domain, NEWLINENEWLINENEWLINE(b) if dim \(R\geq 2\), then \(R\) is an STP domain iff \(R\) has exactly one maximal ideal \(M\) of height greater than one, \(R_M\) is an STP valuation domain and all other maximal ideals have height one and are invertible.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].