In this paper, we introduce and study a new concept (up to our knowledge) named CL-duo modules, which is bigger than that of duo modules, and smaller than weak duo module which is given by Ozcan and Harmanci. Several properties are investigated. Also we consider some characterizations of CL-duo modules. Moreover, many relationships are given for this class of modules with other related classes of modules such as weak duo modules, P-duo modules.

  In this paper we introduced the concept of 2-pure submodules as a generalization of pure submodules, we study some of its basic properties and by using this concept we define the class of 2-regular modules, where an R-module M is called 2-regular module if every submodule is 2-pure submodule. Many results about this concept are given. 

The main goal of this paper is to show that a
-arc in
and
is subset of a twisted cubic, that is, a normal rational curve. The maximum size of an arc in a projective space or equivalently the maximum length of a maximum distance separable linear code are classified. It is then shown that this maximum is
for all dimensions up to
.

      Fuchs introduced purely extending modules as a generalization of extending modules. Ahmed and Abbas gave another generalization for extending modules named semi-extending modules. In this paper, two generalizations of the extending modules are combined to give another generalization. This generalization is said to be almost semi-extending. In fact, the purely extending modules lies between the extending and almost semi-extending modules. We also show that an almost semi-extending module is a proper generalization of purely extending. In addition, various examples and important properties of this class of modules are given and considered. Another characterization of almost semi-extending modules is established. Moreover, the re

     The -s-extending modules will be purpose of this paper, a module M  is -s-extending if each submodule in M is essential in submodule has a supplement that is direct summand. Initially, we give relation between this concept with weakly supplement extending modules and -supplemented modules. In fact, we gives the following implications:

Lifting modules   -supplemented modules   -s-extending modules  weakly supplement extending modules.

It is also we give examples show that, the converse of this result is not true. Moreover, we study when the converse of this result is true.

    In this paper, we formulate and study a new property, namely indeterminacy (neutrosophic) of the hollow module. We mean indeterminacy hollow module is neutrosophic hollow module B (shortly Ne(B)) such that it is not possible to specify the conditions for satisfying it.  Some concepts have been studied and introduced, for instance, the indeterminacy local module, indeterminacy divisible module, indeterminacy indecomposable module and indeterminacy hollow-lifting module. Also, we investigate that if Ne(B) is an indeterminacy divisible module with no indeterminacy zero divisors, then any indeterminacy submodule Ne(K) of Ne(B) is an indeterminacy hollow module. Further, we study the relationship between the indeterminacy of hollow an

     In this work, we introduce a new generalization of both Rationally extending and Goldie extending which is Goldie Rationally extending module which is known as follows: if for any submodule K of an R-module M there is a direct summand U of M (denoted by  U⊆_⊕ M) such that K β_r  U. A β_r  is a relation of K⊆M and U⊆M, which defined as  K β_r  U if and only if  K ⋂U⊆_r K and K⋂U⊆_r U.

       A submodule N of a module M  is said to be s-essential if it has nonzero intersection with any nonzero small submodule in M. In this article, we introduce and study a class of modules in which all its nonzero endomorphisms have non-s-essential kernels, named, strongly -nonsigular. We investigate some properties of strongly -nonsigular modules. Direct summand, direct sums and some connections of such modules are discussed.        

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.