t-Class Semigroups of Integral Domains

Details t-Class Semigroups of Integral Domains t-Class Semigroups of Integral Domains

2006-06-25

arxiv.org/abs/math/0606636v5

J. Reine Angew. Math. 612 (2007) 213-229

Mathematics

Commutative Algebra

Section 4 was removed. J. Reine Angew. Math. (to appear), 17 pages

Scientific paper

The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by ideal t-multiplication. This paper investigates ring-theoretic properties of an integral domain that reflect reciprocally in the Clifford or Boolean property of its t-class semigroup. Contexts (including Lipman and Sally-Vasconcelos stability) that suit best t-multiplication are studied in an attempt to generalize well-known developments on class semigroups. We prove that a Prufer v-multiplication domain (PVMD) is of Krull type (in the sense of Griffin) if and only if its t-class semigroup is Clifford. This extends Bazzoni and Salce's results on valuation domains and Prufer domains. We also characterize GCD domains with Boolean t-class semigroup, recovering thus recent results on Bezout domains.

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