Krull dimension of bimodules

Let R and S be rings and let M be a left S-, right R-bimodule such that M has Krull dimension both as a left S-module and as a right R-module and M/N has the same Krull dimension as a left S- and as a right R-module for every sub-bimodule N of M. Suppose that M is a finitely generated left S-module. Then it is proved that the Krull dimension of the right R-module M is the Krull dimension of a k-critical right R-module M/K for some prime submodule K of the right R-module M. (c) 2007 Elsevier Inc. All rights reserved.