On Cofinitely Injective Modules

In this paper, we introduce the notion of cofinitely injective modules and study their properties over special types of rings. We define a right R ‐module M to be cofinitely injective if it is injective with respect to short exact sequences of right R ‐modules of the form 0→A→B→C→0, where A is cofinitely generated. We completely characterize cofinitely injective modules over a right conoetherian ring, as precisely those modules M , whose torsion part L ( M ) in the simple torsion theory ( L , L ) contains an essential submodule B = → l i m ⁡{ B 1 }1 ∈ Iwhere{ B 1 }1 ∈ Iis a directed system of injective submodules of L ( M ) such that every cofinitely generated submodule of M is contained in some B 1 . This leads to a complete characterization of cofinitely injective modules over a Dedekind domain. Over a right artinian ring, the cofinitely injective modules are precisely the injective modules.