In this work, we introduced and studied  a new kind of soft mapping  on soft topological spaces with an ideal, which we called  soft strongly generalized mapping with respect an ideal I, we studied the concepts like SSIg-continuous, Contra-SSIg-continuous, SSIg-open, SSIg-closed and SSIg-irresolute mapping  and  the relations between these kinds of mappings and the composition of two mappings of the same type of two different types, with proofs or counter examples

  In this paper, we introduce a new class of sets, namely , s*g--open sets and we show that the family of all s*g--open subsets of a topological space ) ,X(  from a topology on X which is finer than  . Also , we study the characterizations and basic properties of s*g-open sets and s*g--closed sets . Moreover, we use these sets to define and study a new class of functions, namely , s*g-  -continuous functions and s*g-  -irresolute functions in topological spaces . Some properties of these functions have been studied .

The aim of this paper is to introduce the concept of N  and Nβ -closed sets in terms of neutrosophic topological spaces. Some of its properties are also discussed.

      In this paper, we introduced module that satisfying strongly -condition modules and strongly -type modules as generalizations of t-extending. A module  is said strongly -condition if for every submodule of  has a complement which is fully invariant direct summand. A module   is said to be strongly -type modules if every t-closed submodule has a complement which is a fully invariant direct summand. Many characterizations for modules with strongly -condition for strongly -type module are given. Also many connections between these types of module and some related types of modules are investigated.

A submoduleA of amodule M is said to be strongly pure , if for each finite subset {ai} in A , (equivalently, for each a ?A) there exists ahomomorphism f : M ?A such that f(ai) = ai, ?i(f(a)=a).A module M is said to be strongly F–regular if each submodule of M is strongly pure .The main purpose of this paper is to develop the properties of strongly F–regular modules and study modules with the property that the intersection of any two strongly pure submodules is strongly pure .