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If $A$ is a domain with the ascending chain condition on (integral) invertible ideals, then the group $I(A)$ of its invertible ideals is generated by the set $I_m(A)$ of maximal invertible ideals. In this note we study some properties of $I_m(A)$ and we prove that, if $I(A)$ is a free group on $I_m(A)$, then $A$ is a locally factorial Krull domain.

We illustrate some results relating the finite character property, the local stability property and the local invertibility property of a domain and give a partial answer to two open questions.

If $A$ is a domain with the ascending chain condition on (integral) invertible ideals, then the group $I(A)$ of its invertible ideals is generated by the set $I_m(A)$ of maximal invertible ideals. In this note we study some properties of $I_m(A)$ and we prove that, if $I(A)$ is a free group on $I_m(A)$, then $A$ is a locally factorial Krull domain.