Noetherian domains which admit only finitely many star operations
Let \(R\) be an integral domain, and let \(S(R)\) be the set of star operations on it. \textit{H. Bass} [Math. Z. 82, 8--28 (1963; Zbl 0112.26604)] proved that \(|S(R)| = 1\) if and only if \(I_v = I\) for each nonzero fractional ideal of \(R\), and if and only if each nonzero fractional ideal is divisorial. \textit{E. Matlis} [J. Algebra 8, 1--33 (1968; Zbl 0191.32301)] showed that a local Noetherian domain \((R, M )\) satisfies \(|S(R)| = 1\) if and only if \(R\) has dimension one and \(M^{ -1}\) is a 2-generated \(R\)-module. For a local Noetherian domain \((R, M )\), the characterization of \(|S(R)| = 2\) is given by the authors in [Commun. Algebra 39, No. 5, 1907--1921 (2011; Zbl 1223.13002)]. In this paper, the authors study Noetherian domains which admit only finitely many star operations and show that, as in the cases \(|S(R)| = 1, 2\), such domains must have Krull dimension one. For a one-dimensional local Noetherian domain \((R, M ), M\) nonprincipal, \(M^{ -1}\) is an overring of \(R\), and \(M^{ -1} /M\) is naturally an \(R/M\) -vector space. They also obtain precise counts of \(S(R)\) in several cases when \(\text{dim}_{R/M} M^{ -1} /M = 3\). Finally, the authors give the answer to the question ``For a local Noetherian domain \(R\), when \(|S(R)| = 3\)?'' under some conditions.
scientific article; zbMATH DE number 6140813
scientific article