In this paper, we introduce the concept of almost cotorsion modules. A module is called almost cotorsion if it is subisomorphic to its cotorsion envelope. Some characterizations of almost cotorsion modules are given. It is also proved that every module is a direct summand of an almost cotorsion module. As an application, perfect rings are characterized in terms of almost cotorsion modules.
Let A and F be artin algebras and ∧UГa paper, we first introduce the notion of k-Gorenstein faithfully balanced selforthogonal bimodule. In this modules with respect to ∧UГ and then characterize it in terms of the U-resolution dimension of some special injective modules and the property of the functors Ext^i （Ext^i （-, U）, U） preserving monomorphisms, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. In addition, we give some properties of ∧UГwith finite left or right injective dimension.
For a ring R, let ip（RR）={a ∈ R： every right R-homomorphism f from any right ideal of R into R with Imf = aR can extend to R}. It is known that R is right IP-injective if and only if R = ip（RR） and R is right simple-injective if and only if {a ∈ R ： aR is simple} ∪→ ip（RR）. In this note, we introduce the concept of right S-IP-injective rings, i.e., the ring R with S ∪→ ip（RR）, where S is a subset of R. Some properties of this kind of rings are obtained.
