On the cardinality of star operations on a pseudo-valuation domain
The author characterizes when a pseudo-valuation domain (\(PVD\) for short) has a finite number of star and semistar operations. The results obtained are summarized as follows: Let \(R\) be a \(PVD\) (which is not a valuation domain), \(V\) its associated valuation overring, \(M\) its maximal ideal, \(k=R/M\) its residue field and \(L=V/M\). Let \(\text{Star}(R)\) and \(\text{SStar}(R)\) be the sets of all star and semistar operations of \(R\) respectively. Then: (1) \(|\text{Star}(R)|<\infty\) if and only if \([L:k]= 2, 3\). (\([L:k]=2\) if and only if \(R\) is divisorial, equivalently \(|\text{Star}(R)|=1\), and if \([L:k]=3\), then \(|\text{Star}(R)|=2\)). (2) \(|\text{SStar}(R)|<\infty\) if and only if \(|\text{Star}(R)|<\infty\) and \(\dim R<\infty\).
scientific article; zbMATH DE number 6152192