WEAKLY-INJECTIVE MODULES OVER HEREDITARY NOETHERIAN PRIME-RINGS

A module M is said to be weakly-injective if and only if for every fin itely generated submodule N of the injective hull E(M) of M there exis ts a submodule X of E(M), isomorphic to M such that N subset of X. In this paper we investigate weakly-injective modules over bounded heredi tary noetherian prime rings. In particular we show that torsion-free m odules over bounded hnp rings are always weakly-injective, while torsi on modules with finite Goldie dimension are weakly-injective only if t hey are injective. As an application, we show that weakly-injective mo dules over bounded Dedekind prime rings have a decomposition as a dire ct sum of an injective module B, and a module C satisfying that if a s imple module S is embeddable in C then the (external) direct sum of al l proper submodules of the injective hull of S is also embeddable in C . Indeed, we show that over a bounded hereditary noetherian prime ring every uniform module has periodicity one if and only if every weakly- injective module has such a decomposition.