In this paper, we introduce two new notions related with coretrectable modules which are called essentially coretrectable and weakly essentially coretrectable modules. Some basic properties of these classes are investigated and some relationships between these modules and other related concepts are given.

In this paper, as generalization of second modules we introduce type of modules namely (essentially second modules). A comprehensive study of this class of modules is given, also many results concerned with this type and other related modules presented.

In this paper we introduce the notions of t-stable extending and strongly t-stable extending modules. We investigate properties and characterizations of each of these concepts. It is shown that a direct sum of t-stable extending modules is t-stable extending while with certain conditions a direct sum of strongly t-stable extending is strongly t-stable extending. Also, it is proved that under certain condition, a stable submodule of t-stable extending (strongly t-stable extending) inherits the property.

In this paper, we introduce a type of modules, namely S-K-nonsingular modules, which is a generalization of K-nonsingular modules. A comprehensive study of these classes of modules is given.

In this paper , we define the concept of a smarandache anti fuzzy M-semigroup (S-Anti Fuzzy M-Semigroup) and some elementary properties about this concept are discussed.