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

Abstract

Let [Formula: see text] be a positively graded ring which is a sub-ring of strongly graded ring of type [Formula: see text], where [Formula: see text] is a Noetherian prime ring. We define a concept of [Formula: see text]-invariant maximal order and show that [Formula: see text] is a maximal order if and only if [Formula: see text] is a [Formula: see text]-invariant maximal order. If [Formula: see text] is a maximal order, then we completely describe all [Formula: see text]-invertible ideals. As an application, we show that [Formula: see text] is a generalized Dedekind prime ring if and only if [Formula: see text] is a [Formula: see text]-invariant generalized Dedekind prime ring. We give examples of [Formula: see text]-invariant generalized Dedekind prime rings but neither generalized Dedekind prime rings nor maximal orders.