On strong indecomposability of the Dedekind ring localization

In this work, we study the question of sufficient conditions for strong indecomposability of Dedekind rings as finite rank modules over their Dedekind subrings. This question was considered earlier in the special case of rings of algebraic integers. Necessary and sufficient conditions are given for Galois extensions. The question of $k$-indecomposability is also considered.