INTEGRAL DOMAINS WITH A FREE SEMIGROUP OF *-INVERTIBLE INTEGRAL *-IDEALS

Abstract

Let * be a star-operation on an integral domain R, and let <TEX>$\mathfrak{I}_*^+(R)$</TEX> be the semigroup of *-invertible integral *-ideals of R. In this article, we introduce the concept of a *-coatom, and we then characterize when <TEX>$\mathfrak{I}_*^+(R)$</TEX> is a free semigroup with a set of free generators consisting of *-coatoms. In particular, we show that <TEX>$\mathfrak{I}_*^+(R)$</TEX> is a free semigroup if and only if R is a Krull domain and each <TEX>${\upsilon}$</TEX>-invertible <TEX>${\upsilon}$</TEX>-ideal is *-invertible. As a corollary, we obtain some characterizations of *-Dedekind domains.