On equivalence of graded rings (Q1380961)

Let \(G\), \(H\) be groups, \(R\) a \(G\)-graded ring, and \(S\) an \(H\)-graded ring. Then \(R\) and \(S\) are called homogeneously equivalent if there exists a ring isomorphism \(f\colon R\to S\) such that \(f(h(R))=h(S)\), where \(h(R)\) denotes the set of homogeneous elements of \(R\). In this case, the connection between graded ideals of \(R\) and \(S\) is studied, and it is proved that the following properties are inherited by homogeneous equivalence: graded semiprime, graded prime, graded noetherian (artinian), graded simple. The relationship between strongly graded type conditions for \(R\) and \(S\) is studied, and some applications for \(\mathbb{Z}\)-graded rings are given.