The class semigroup of local one-dimensional domains
In recent years, several papers have investigated the question of when the class semigroup, derived from the classical definition of the class group, of a commutative integral domain is a Clifford semigroup. This paper investigates conditions for the class semigroup \(S(R)\) of a local one-dimensional domain \((R, \mathfrak m)\) to be Clifford. Recall that a semigroup \(S\) is called Clifford if it satisfies one of the three equivalent conditions: (1) every element of \(S\) is contained in a subgroup \(G\) of \(S\); (2) for every \(x \in S\) there is \(y \in S\) such that \(x = xyx\); (3) \(S\) is a semilattice of groups. Because \textit{S. Bazzoni} [J. Algebra 238, 703--722 (2001; Zbl 1027.13011)] proved that an integrally closed domain has a Clifford class semigroup if and only if the ring is a Prüfer domain of finite character, the author focuses on the case when the ring is not integrally closed. He makes use of Archimedean valuation domains which dominate \(R\). To be specific, he shows that if \(R\) is dominated by a unique (resp., two distinct) Archimedean valuation domain(s) \(V\) (resp., \(V_1\) and \(V_2\)), and if \(S(R)\) is a Clifford semigroup, then \(V\) (resp., \(V_1 \cap V_2\)) coincides with the integral closure \(\overline{R}\) of \(R\), and every element of \(\overline{R}\) has degree less than or equal to two over \(R\). On the other hand, if any of the following conditions hold, then \(S(R)\) is not Clifford: (i) \(R\) is dominated by three distinct Archimedean valuation domains; (ii) \(R\) is dominated by a nondiscrete real valuation \(v\) and \(v(\mathfrak m) >0\); (iii) \(v\) is a real valuation dominating \(R\) and \(v(\mathfrak m) \geq 3\); (iv) \(v\) is a real valuation, dominating \(R\), of a valuation domain \((V, P)\) such that \([V/P:R/\mathfrak m] = 2\) and \(v(\mathfrak m) =2\). When \(R\) contains a nonzero ideal of a DVR \(V\), the author completely determines when \(S(R)\) is Clifford. Another main result of the paper is the following: Let \(R\) be a local one dimensional domain such that \(S(R)\) is a Clifford semigroup. Then \(I \cong I^2\) for every finitely generated ideal \(I\) of \(R\). Moreover, if the real valuations dominating \(R\) are all discrete, then \(S(R)\) is a Boolean semigroup. Throughout this work, the author provides several examples.
scientific article; zbMATH DE number 5299442
scientific article