On Noetherian modules graded by \(G\)-sets (Q1912662)

Let \(G\) be a group, \(R\) a \(G\)-graded ring, \(A\) a left \(G\)-set, and \(M\) an \(R\)-module graded by \(A\). The paper deals with the following problem: if \(M\) is noetherian as a graded module, does it follow that \(M\) is noetherian? The following cases are discussed: 1. \(G\) is finite, \(A\) is arbitrary. 2. \(G\) is abelian, \(A\) is finite. 3. \(G=\mathbb{Z}\times F\), \(F\) is finite abelian, \(A\) is countable.