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 R be a ring and S a class of R-modules. S-superfluous epimorphisms and S-essential monomorphisms are introduced and studied in this article. As applications, some new characterizations of von Neumann regular rings and perfect rings are given. Finally, these notions are also used to study minimal homomorphisms.
