176,665 research outputs found
On --domains and star operations
Let be a star operation on an integral domain . Let \f(D) be the
set of all nonzero finitely generated fractional ideals of . Call a
--Pr\"ufer (respectively, --Pr\"ufer) domain if
(respectively, ) for all F\in
\f(D). We establish that --Pr\"ufer domains (and --Pr\"ufer
domains) for various star operations span a major portion of the known
generalizations of Pr\"{u}fer domains inside the class of --domains. We also
use Theorem 6.6 of the Larsen and McCarthy book [Multiplicative Theory of
Ideals, Academic Press, New York--London, 1971], which gives several equivalent
conditions for an integral domain to be a Pr\"ufer domain, as a model, and we
show which statements of that theorem on Pr\"ufer domains can be generalized in
a natural way and proved for --Pr\"ufer domains, and which cannot be. We
also show that in a --Pr\"ufer domain, each pair of -invertible
-ideals admits a GCD in the set of -invertible -ideals,
obtaining a remarkable generalization of a property holding for the "classical"
class of Pr\"ufer --multiplication domains. We also link being --Pr\"ufer (or --Pr\"ufer) with the group Inv of -invertible -ideals (under -multiplication) being
lattice-ordered
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