Throughout this paper R represents commutative ring with identity and M is a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of weak essential submodules which introduced by Muna A. Ahmed, where a submodule N of an R-module M is called weak essential, if N ? P ? (0) for each nonzero semiprime submodule P of M. In this paper we rewrite this definition in another formula. Some new definitions are introduced and various properties of weak essential submodules are considered.

The family Ormyridae has been very much neglected by workers and only two species has been recorded so far from Iraq. The present study, based mainly on my collection, deals with five species, of which one is new to science. The new species is described together with notes on locality data, host records, distribution and taxonomical remarks for all the species.

Czerwi’nski et al. introduced Lucky labeling in 2009 and Akbari et al and A.Nellai Murugan et al studied it further. Czerwi’nski defined Lucky Number of graph as follows: A labeling of vertices of a graph G is called a Lucky labeling if  for every pair of adjacent vertices u and v in G where . A graph G may admit any number of lucky labelings. The least integer k for which a graph G has a lucky labeling from the set 1, 2, k is the lucky number of G denoted by η(G). This paper aims to determine the lucky number of Complete graph K_n, Complete bipartite graph K_m,n and Complete tripartite graph K_l,m,n. It has also been studied how the lucky number changes whi

  Let â„› be a commutative ring with unity and let ℬ be a unitary R-module. Let ℵ be a proper submodule of ℬ, ℵ is called semisecond submodule if for any r∈ℛ, r≠0, n∈Z_+, either rⁿℵ=0 or rⁿℵ=rℵ.

In this work, we introduce the concept of semisecond submodule and confer numerous properties concerning with this notion. Also we study semisecond modules as a popularization of second modules, where an ℛ-module ℬ is called semisecond, if ℬ is semisecond submodul of ℬ.

In this paper we give many connections between essentially quasi-Dedekind (quasi-
Dedekind) modules and other modules such that Baer modules, retractable modules,
essentially retractable modules, compressible modules and essentially compressible
modules where an R-module M is called essentially quasi-Dedekind (resp. quasi-
Dedekind) if, Hom(M N ,M )  0 for all N ≤e M (resp. N ≤ M). Equivalently, a
module M is essentially quasi-Dedekind (resp. quasi-Dedekind) if, for each
f End (M) R  , Kerf ≤ e M implies f = 0 (resp. f  0 implies ker f  0 ).

Reacts compound C6H5PO2Cl2 with Secretary secondary R2NH at room temperature by Mulet 2:1 and using chloroform as a solvent in dry conditions to form composite 2HCl and the interaction of compound solution of sodium hydroxide and potassium by Mulet 3:1 salt was prepared

 Let  be a connected graph with vertices set  and edges set . The ordinary distance between any two vertices of  is a mapping  from  into a nonnegative integer number such that  is the length of a shortest  path. The maximum distance between two subsets  and  of   is the maximum distance between any two vertices  and  such that  belong to  and  belong to . In this paper, we take a special case of maximum distance when  consists of one vertex and  consists of  vertices, . This distance is defined by: where  is the order of  a graph .

     In this paper, we defined  – polynomials based on

Throughout this paper R represents commutative ring with identity and M is a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of weak essential submodules which introduced by Muna A. Ahmed, where a submodule N of an R-module M is called weak essential, if N ? P ? (0) for each nonzero semiprime submodule P of M. In this paper we rewrite this definition in another formula. Some new definitions are introduced and various properties of weak essential submodules are considered.

    The definition of semi-preopen sets were first introduced by "Andrijevic" as were is defined by :Let (X ,  ) be a topological space, and let  A ⊆,    then Ais called semi-preopen set if ⊆∘ .        In this paper, we study the properties of semi-preopen sets but by another  definition which is equivalent to the first definition and we also study the relationships among it and (open, α-open, preopen and semi-p-open )sets.