Class Semigroups of Integral Domains
Abstract
This paper seeks ringtheoretic conditions of an integral domain R that reflect in the Clifford property or Boolean property of its class semigroup S(R), that is, the semigroup of the isomorphy classes of the nonzero (integral) ideals of R with the operation induced by multiplication. Precisely, in Section 3, we characterize integrally closed domains with Boolean class semigoup; in this case, S(R) identifies with the Boolean semigroup formed of all fractional overrings of R. In Section 4, we investigate Noetherianlike settings where the Clifford and Boolean properties of S(R) coincide with (Lipman and SallyVasconcelos) stability conditions; a main feature is that the Clifford property forces tlocally Noetherian domains to be onedimensional Noetherian domains. Section 5 studies the transfer of the Clifford and Boolean properties to various pullback constructions. Our results lead to new families of integral domains with Clifford or Boolean class semigroup, moving therefore beyond the contexts of integrally closed domains or Noetherian domains.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606691
 Bibcode:
 2006math......6691K
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Number Theory;
 13C20;
 13F05;
 13F30;
 13A15;
 13B22;
 13G05;
 20M14;
 11R29
 EPrint:
 19 pages