Modules over crossed products
Let \(R\) be either a division ring or a left Noetherian simple ring of Krull dimension 1. Suppose that \(G\) is a supersolvable group and \(C=R*G\) a crossed product. Assume that \(C\) is a simple right Ore domain. Then any left ideal of \(C\) is stably 2-generated, any left stably free module of rank \(\geq 2\) is free. If \(M\) is a finitely generated left \(C\)-module then \(M=N\oplus C^d\) where \(\text{rank }N\leq 1\). If \(M\) is torsion-free then \(N\) is isomorphic to a left ideal of \(C\). A finitely generated torsion \(C\)-module is isomorphic to \(I/J\) where \(I\) is a left projective ideal of \(C\). These results generalize the reviewer's results for the case of general quantum Laurent polynomials [from Mat. Zametki 59, No. 4, 497-503 (1996; Zbl 0879.16013), ibid. 61, No. 1, 10-17 (1997; Zbl 0924.16006)]. Some corrections to the first paper are indicated in the reviewer's paper [Usp. Mat. Nauk 53, No. 4, 3-76 (1998)]. Similar results are proved for a domain \(A\) which is a tensor product of simple generalized Weyl algebras in the sense of \textit{V. V. Bavula} [Ukr. Mat. Zh. 44, No. 12, 1628-1644 (1992; Zbl 0810.16003)].
scientific article; zbMATH DE number 1295453