Strongly graded rings over hereditary Noetherian prime rings (Q1988543)

Let \(R\) be a strongly \(\mathbb{Z}\)-graded ring such that the homogeneous component \(R_0\) of trivial degree is a hereditary Noetherian prime ring. The authors investigate maximal \(\mathbb{Z}\)-invariant ideals of \(R_0\), and they show that each such ideal is either idempotent or invertible. It is also proved that the invertible \(\mathbb{Z}\)-invariant ideals of \(R_0\) generate an abelian group whose generators are the maximal invertible \(\mathbb{Z}\)-invariant ideals of \(R_0\). These results are used to uncover the structure of projective ideals of \(R\). As an application, it is showed that \(R\) is a strongly G-HNP ring, a strong version of a property defined by the authors in [J. Algebra Appl. 17, No. 8, Article ID 1850153, 22 p. (2018; Zbl 1414.16008)].