Unique decomposition into ideals for reduced commutative Noetherian rings, II

Let R be a reduced commutative Noetherian ring. We provide conditions equivalent to isomorphism for completely decomposable finitely generated modules over R. We show that, if R is one dimensional and R satisfies the Krull–Schmidt property for ideals, then any overring of R must also have this property. We also show that if R is both local and one dimensional, satisfying the Krull–Schmidt property for ideals, then it has the Krull–Schmidt property for direct sums of rank one modules.