On one class of modules that are close to Noetherian

We consider an RG-module A over a commutative Noetherian ring R. Let G be a group having infinite section p-rank (or infinite 0-rank) such that CG(A) = 1, A/CA(G) is not a Noetherian R-module, but the quotient A/CA(H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (or infinite 0-rank, respectively). In this paper, it is proved that if G is a locally soluble group, then G is soluble. Some properties of soluble groups of this type are also obtained. © 2010 Springer Science+Business Media, Inc.