Class semigroups of integral domains
Let \(R\) be an integral domain with quotient field \(K\). The class semigroup \({\mathcal S}(R)\) of \(R\) is the monoid of the nonzero fractional ideals of \(R\) under multiplication modulo its subgroup of nonzero principal fractional ideals. Recall that a semigroup \(S\) is a Clifford (respectively Boolean) semigroup if for each \(x\in S\), there exists an \(a\in S\) with \(x^2a=x\) (respectively \(x^2=x\)). \textit{P. Zanardo} and \textit{U. Zannier} [Math. Proc. Camb. Philos. Soc. 115, 379-391 (1994; Zbl 0828.11068)] showed that an integrally closed domain \(R\) with \({\mathcal S}(R)\) a Clifford semigroup is a Prüfer domain, but that the converse is false. Then \textit{S. Bazzoni} [``Clifford regular domains'', J. Algebra 238, 703-722 (2001; see the preceding review Zbl 1027.13011)] showed that an integrally closed domain \(R\) has \({\mathcal S}(R)\) a Clifford semigroup iff \(R\) is a Prüfer domain of finite character. The main result of the paper under review is that an integrally closed domain \(R\) has \({\mathcal S}(R)\) a Boolean semigroup iff \(R\) is a strongly discrete Bezout domain of finite character iff each nonzero ideal of \(R\) is principal in its endomorphism ring. It is also shown that for \(R\) a \(t\)-locally Noetherian domain, \(R\) has Clifford (respectively Boolean) class semigroup iff \(R\) is stable (respectively each nonzero ideal of \(R\) is principal in its endomorphism ring). The transfer of the Clifford and Boolean properties to pullbacks is also discussed.
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