Bulletin of the
Korean Mathematical Society BKMS

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