Locally pseudo-valuation domains with only finitely many star operations
The overall goal of this article is very similar to [\textit{E. Houston} et al., Commun. Algebra 42, No. 12, 5264--5286 (2014; Zbl 1310.13030)], where integrally closed domains with finitely many star operations were characterized. Here, the authors turn their attention to two global counterparts of pseudo-valuation domains and seek to similarly characterize which of these domains have finitely many star operations. Let \(R\) be a domain with quotient field \(K\). Then \(R\) is a pseudo-valuation domain (PVD) if it is quasi-local with a maximal ideal \(M\) such that \((M:M)\) is a valuation domain with maximal ideal \(M\). There are two global counterparts to PVDs introduced in [\textit{D. E. Dobbs} and \textit{M. Fontana}, Ann. Mat. Pura Appl. (4) 134, 147--168 (1983; Zbl 0531.13012)]. The first is a locally pseudo-valuation domain (LPVD) which is a domain \(R\) such that each localization of \(R\) at a maximal ideal is a PVD. The second is a globalized pseudo-valuation domain (GPVD) which is a domain \(R\) which is a subring of a Prüfer domain \(T\) such that the extension \(R\subseteq T\) is unibranched and there is a non-zero common radical ideal \(A\) of \(R\) and \(T\) such that each prime ideal of \(T\) (resp. \(R\)) that contains \(A\) is a maximal ideal of \(T\) (resp. \(R\)). The authors show that an LPVD \(R\) has only finitely many star operations if and only if \(R\) is ``nearly'' \(h\)-local, the PVD \(R_M\) has only finitely many star operations for each maximal ideal \(M\) of \(R\), and \(R_M\) is divisorial for almost all maximal ideals \(M\). They also show that if \((R:T)\neq (0)\): \(R\), then a GPVD has only finitely many star operations if an only if \(T\) and each \(R_M\) have only finitely many star operations. On the other hand, when \((R:T) = (0)\), the authors demonstrate how neither of the previous implications hold by providing specific examples which show that all of the various possibilities can arise.
scientific article; zbMATH DE number 6493617