Ideal class semigroups of integral domains
The ideal class semigroup of an integral domain R consists of all isomorphism classes of nonzero fractional ideals of R, under usual multiplication. 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. We investigate conditions under which this semigroup homomorphism is surjective and we apply the results we obtain to the study of overrings of Clifford regular domains, where a domain is Clifford regular if its ideal class semigroup is a disjoint union of abelian groups. We recover some known results of Bazzoni and we prove in certain more general situations that the Clifford regular property is inherited by an overring. In particular, we prove 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. We also characterize among Clifford regular domains the ones that are stable. For a local domain R with idempotent archimedean maximal ideal, Bazzoni asked whether R Clifford regular or finitely stable implies that R is a valuation domain. We prove that if R is Clifford regular, the answer is yes; our proof is based on the fact that idempotent integral ideals of Clifford regular domains are radical ideals. If R is only finitely stable we prove the answer is no, in general. In particular, we establish the existence of a finitely stable domain having a nonradical idempotent ideal. We also study the behavior of Clifford regularity under integral extensions. The concept of a domain of finite character plays an important role in our investigation. We describe the domains all of whose overrings have finite character. We prove that the integral closure of a semilocal Clifford regular domain is again Clifford regular. We study the integral closure of a Clifford regular domain in an extension of its fraction field.
Heinzer, Purdue University.
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