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.
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机译：假设$ * $是整数域$ R $上的星型运算，并且让$ mathscr {I} _ {*} ^ {+}（R）$是$ * $可逆整数$ * $的半群-$ R $的理想选择。在本文中，我们介绍了$ * $-coatom的概念，然后描述了$ mathscr {I} _ {*} ^ {+}（R）$是一个带有一组免费生成器的免费半群的情况， $ * $-coatoms。特别是，我们证明，当且仅当$ R $是一个Krull域并且每个$ v $可逆$ v $-时，$ mathscr {I} _ {*} ^ {+}（R）$是一个自由半群。理想的是$ * $可逆。作为推论，我们获得了$ * $-Dedekind域的一些特征。
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