Positively graded rings which are maximal orders and generalized Dedekind prime rings

Let R=⊕n∈Z0Rn be a positively graded ring which is a sub-ring of strongly graded ring of type Z, where R0 is a Noetherian prime ring. We define a concept of Z0-invariant maximal order and show that R is a maximal order if and only if R0 is a Z0-invariant maximal order. If R is a maximal order, then we completely describe all v-invertible ideals. As an application, we show that R is a generalized Dedekind prime ring if and only if R0 is a Z0-invariant generalized Dedekind prime ring. We give examples of Z0-invariant generalized Dedekind prime rings but neither generalized Dedekind prime rings nor maximal orders.