Kaplansky in Theorem 8 of This paper state that, "any module $M$ over a Dedekind ring possesses a unique largest divisibel submodule $D$; $M=D\oplus E$ where $E$ has no divisible submodules."
Is there weaker conditions (than Dedekind domain which is very strong) on a ring such that this Theorem be still true? Thanks a lot.