Strongly graded rings which are generalized Dedekind rings

Let R=⊕n∈ZRn be a strongly graded ring of type Z such that R0 is a prime Goldie ring with its quotient ring Q0. It is shown that the following three conditions are equivalent: (i) R0 is a Z-invariant generalized Dedekind ring (G-Dedekind ring for short), (ii) R is a G-Dedekind ring and (iii) R is a graded G-Dedekind ring. We describe all invertible ideals of G-Dedekind rings in terms of R0 and Qg=⊕n∈ZQ0Rn. We provide counterexamples of Z-invariant G-Dedekind rings which are not G-Dedekind rings.