Let A, and N  are a semiring ,and  a left A- semimodule, respectively. In this work we will discuss two cases: 

1. The direct summand of π-projective semi module is π-projective, while the direct sum of two π-projective semimodules in general is not π-projective . The details of the proof will be given.
2. We will give a condition under which the direct sum of two π-projective semi modules is π-projective, as well as we also set conditions under which π-projective semi modules are projective.

       The basis of this paper is to study the concept of almost projective semimodules as a generalization of projective semimodules. Some of its characteristics have been discussed, as well as some results have been generalized from projective semimodules.

     In this work,  injective semimodule  has been generalized to  almost -injective semimodule. The aim of this research is  to study the basic properties of the concept almost- injective semimodules. The semimodule  is called  almost  -injective semimodule if, for each subsemimodule A of  and each homomorphism  : A   , either there exists  a homomorphism  such that = . Or there exists a homomorphism : Y such that = , where Y is nonzero direct summand of , and   is the  projection map. A semimodule  is almost injective semimodule if it is almost injective relative to all semimodules. Every injective semimodule is almost injective semimodule,  if  is almost  –

In this paper, we study the class of prime semimodules and the related concepts, such as the class of  semimodules, the class of Dedekind semidomains, the class of prime semimodules which is invariant subsemimodules of its injective hull, and the compressible semimodules. In order to make the work as complete as possible, we stated, and sometimes proved, some known results related to the above concepts.

In this paper we study the notion of preradical on some subcategories of the category of semimodules and homomorphisms of semimodules.
Since some of the known preradicals on modules fail to satisfy the conditions of preradicals, if the category of modules was extended to semimodules, it is necessary to investigate some subcategories of semimodules, like the category of subtractive semimodules with homomorphisms and the category of subtractive semimodules with ҽҟ-regular homomorphisms.

        The purpose of this work is to construct complete (k,n)-arcs in the projective 2-space PG(2,q) over Galois field GF(11) by adding some points of index zero to complete (k,n–1)arcs 3 ï‚£ n ï‚£ 11.         A (k,n)-arcs is a set of k points no n + 1 of which are collinear.         A (k,n)-arcs is complete if it is not contained in a (k + 1,n)-arc

        Let R be an associative ring. In this paper we present the definition of (s,t)- Strongly derivation pair and Jordan (s,t)- strongly derivation pair on a ring R, and study the relation between them. Also, we study prime rings, semiprime rings, and rings that have commutator left nonzero divisior with (s,t)- strongly derivation pair, to obtain a (s,t)- derivation. Where s,t: R®R are two mappings of R.

 In this work, we introduced the Jacobson radical (shortly Rad (Ș)) of the endomorphism semiring Ș =  ( ), provided that  is principal P.Q.- injective semimodule and some related concepts, we studied some properties and added conditions that we needed. The most prominent result is obtained in section three

-If   is a principal self-generator semimodule, then (_ȘȘ) = W(Ș).

Subject Classification: 16y60

The aim of this paper is to introduce the concept of Dedekind semimodules and study the related concepts, such as the class of  semimodules, and Dedekind multiplication semimodules .  And thus study the concept of the embedding of a semimodule in another semimodule. 

  Necessary and sufficient conditions for the operator equation I AXAX n  * , to have a real positive definite solution X are given. Based on these conditions, some properties of the operator A as well as relation between the solutions X andAare given.