Jacobson's conjecture and modules over fully bounded Noetherian rings

Abstract

The object of this paper is to investigate finitely generated modules and injective modules over fully bounded Noetherian rings. Our main results on f.g. modules (Theorems 3.1 and 3.4) provide an analog of the Jordan-Holder Theorem and seem to be new even for commutative Noetherian rings. They imply the validity of Jacobson's conjecture for fully bounded Noetherian rings. The main result on injectives (Theorem 5.3) describes f.g. submodules of an indecomposible injective and shows how it can be built from indecomposible injectives over certain Artinian rings.