Ideal class semigroups of overrings
The ideal class semigroup of an integral domain \(R\), denoted by \(S(R)\), consists of all isomorphism classes of nonzero fractional ideals of \(R \), under usual multiplication. An integral domain \(R\) is called Clifford regular if the ideal class semigroup \(S(R)\) is a Clifford semigroup, i.e., for each nonzero fractional ideal \(I\) of \(R\) there exists a nonzero fractional ideal \(J\) of \(R\) such that \(I = I^2 J\). Clifford regular domains were introduced by \textit{P. Zanardo} and \textit{U. Zannier} [Math. Proc. Camb. Phil. Soc. 115, 379--391 (1994; Zbl 0828.11068)] and \textit{S. Bazzoni} and \textit{L. Salce} [Isr. J. Math. 95, 135--155 (1996; Zbl 0869.13006)]. If \(T\) is an overring of \(R \), there is a canonical semigroup homomorphism between the ideal class semigroup of \(R\) and the ideal class semigroup of \(T\). In the present paper, the author investigates conditions under which this semigroup homomorphism is surjective, in order to apply this information to the study of overrings of Clifford regular domains. Following Olberding's terminology, an integral domain \(R\) is called stable (respectively, finitely stable) if each nonzero (respectively, nonzero finitely generated) fractional ideal \(I\) of \(R\) is invertible in the overring \((I:I)\). \textit{S. Bazzoni} [J. Algebra 238, 703--722 (2001; Zbl 1027.13011)] proved that the class of Clifford regular domains is properly intermediate between the class of stable domains and the class of finitely stable domains; in particular, a noetherian domain is Clifford regular if and only if it is a stable domain. After recovering some known results of Bazzoni, the author proves in certain more general situations that the Clifford regular property is inherited by an overring. In particular, he shows that if \(R\) is a Clifford regular domain such that the integral closure of \( R\) is a fractional overring, then every overring of \(R\) is Clifford regular. He also characterizes among Clifford regular domains the ones that are stable and, for local domains whose integral closure is not a valuation domain, he shows that the Clifford regularity is inherited by every overring.
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