Let R be a commutative ring with identity and let M be a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of semi-essential submodules which introduced by Ali S. Mijbass and Nada K. Abdullah, and we make simple changes to the definition relate with the zero submodule, so we say that a submodule N of an R-module M is called semi-essential, if whenever N ∩ P = (0), then P = (0) for each prime submodule P of M. Various properties of semi-essential submodules are considered.

    Let  be a non-zero right module over a ring  with identity. The weakly second submodules is studied in this paper. A non-zero submodule  of   is weakly second Submodule when  ,  where ,  and  is a submodule of  implies either  or   . Some connections between these modules and other related modules are investigated and number of conclusions  and characterizations are gained.

    This paper devoted to the analysis of regular singular initial value problems for ordinary differential equations with a singularity of the first kind , we propose semi - analytic technique using two point osculatory interpolation to construct polynomial solution, and discussion behavior of the solution in the neighborhood of the regular singular points and its numerical approximation, two examples are presented to demonstrate the applicability and efficiency of the methods. Finally , we discuss behavior of the solution in the neighborhood of the singularity point which appears to perform satisfactorily for singular problems.

         Let R be a commutative ring , the pseudo – von neuman  regular graph of the ring R is define as a graph whose vertex set consists of all elements of R and any two distinct vertices a and b are adjacent if and only if   , this graph denoted by P-VG(R) ,  in this work we got some new results a bout chromatic number of  P-VG(R).

We introduce the notion of t-polyform modules. The class of t- polyform modules contains the class of polyform modules and contains the class of t-essential quasi-Dedekind.

     Many characterizations of t-polyform modules are given. Also many connections between these class of modules and other types of modules are introduced.

Throughout this paper, T is a ring with identity and F is a unitary left module over T. This paper study the relation between semihollow-lifting modules and semiprojective covers. proposition 5 shows that If T is semihollow-lifting, then every semilocal T-module has semiprojective cover. Also, give a condition under which a quotient of a semihollow-lifting module having a semiprojective cover. proposition 2 shows that if K is a projective module. K is semihollow-lifting if and only if For every submodule A of K with K/( A) is hollow, then K/( A) has a semiprojective cover.

Let
be an
module,
be a fuzzy soft module over
, and
be a fuzzy soft ring over
, then
is called FSFS module if and only if
is an
module. In this paper, we introduce the concept of
Noetherian and
Artinian modules and finally we investigate some basic properties of
Noetherian and
Artinian modules.

The main purpose of this work is to introduce the concept of higher N-derivation and study this concept into 2-torsion free prime ring we proved that:Let R be a prime ring of char. 2, U be a Jordan ideal of R and be a higher N-derivation of R, then , for all u U , r R , n N .

The idea of a homomorphism of a cubic set of a KU-semigroup is studied and the concept of the product between two cubic sets is defined. And then, a new cubic bipolar fuzzy set in this structure is discussed, and some important results are achieved. Also, the product of cubic subsets is discussed and some theorems are proved. 2010 AMS Classification: 06F35, 03G25, 08A72.