Modules over Crossed Products

J. T. Stafford (1978,J. London Math. Soc. (2)18, 429–442) proved that any left ideal of the Weyl algebraAn(K) over a fieldKof characteristic 0 can be generated by two elements. In general, there is the problem of determining whether any left ideal of a Noetherian simple domain can be generated by two elements. In this work we show that this property holds for some crossed products of a simple ring with a supersolvable group and also for the tensor product of generalized Weyl algebras. We also prove that these rings are stably generated by 2 elements and that their finitely generated torsion left modules can be generated by two elements. Some results about stably 2-generated rings were found by V. A. Artamonov (1994,Math. Sb.185, No. 7, 3–12).