Integral domains with a free semigroup of $*$-invertible integral $*$-ideals
Bull. Korean Math. Soc. 2011 Vol. 48, No. 6, 1207-1218
https://doi.org/10.4134/BKMS.2011.48.6.1207
Published online November 1, 2011
Gyu Whan Chang and Hwankoo Kim
University of Incheon, Hoseo University
Abstract : Let $*$ be a star-operation on an integral domain $R$, and let $\mathscr{I}_{*}^{+}(R)$ be the semigroup of $*$-invertible integral $*$-ideals of $R$. In this article, we introduce the concept of a $*$-coatom, and we then characterize when $\mathscr{I}_{*}^{+}(R)$ is a free semigroup with a set of free generators consisting of $*$-coatoms. In particular, we show that $\mathscr{I}_{*}^{+}(R)$ is a free semigroup if and only if $R$ is a Krull domain and each $v$-invertible $v$-ideal is $*$-invertible. As a corollary, we obtain some characterizations of $*$-Dedekind domains.
Keywords : star-operation, free semigroup of $*$-invertible $*$-ideals, $*$-locally factorial Krull domain, $\pi$-domain, $*$-Dedekind domain
MSC numbers : 13A15, 13F15, 13E99
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