The main purpose of this paper is to investigate some results. When h is  -( ,δ) – Derivation on prime Γ-near-ring G and K is a nonzero semi-group ideal of G, then G is commutative .

In this paper, the concept of Jordan triple higher -homomorphisms on prime

rings is introduced.  A result of Herstein is extended on this concept from the ring  into the prime ring .  We prove that every Jordan triple higher -homomorphism of ring  into prime ring  is either triple higher -homomorphism  or triple higher -anti-homomorphism of  into .

In this paper,  we introduce the concepts of  higher reverse left (resp.right)   centralizer, Jordan higher reverse left (resp. right) centralizer, and Jordan triple higher reverse left (resp. right) centralizer of  G-rings. We prove that every Jordan higher reverse left (resp. right) centralizer of a 2-torsion free prime G-ring M is a higher reverse left (resp. right) centralizer of  M.

   In this paper, a new class of sets, namely - semi-regular closed sets is introduced and studied for topological spaces. This class properly contains the class of semi--closed sets and is property contained in the class of pre-semi-closed sets. Also, we introduce and study srcontinuity and sr-irresoleteness. We showed that sr-continuity falls strictly in between semi-- continuity and pre-semi-continuity.

  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.

  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 .

In this paper, we introduce weak and strong forms of ω-perfect mappings, namely the -ω-perfect, weakly -ω-perfect and strongly-ω-perfect mappings. Also, we investigate the fundamental properties of these mappings. Finally, we focused on studying the relationship between weakly -ω-perfect and strongly -ω-perfect mappings.

In this paper, we study the concepts of generalized reverse derivation, Jordan
generalized reverse derivation and Jordan generalized triple reverse derivation on -
ring M. The aim of this paper is to prove that every Jordan generalized reverse
derivation of -ring M is generalized reverse derivation of M.

This paper consist some new generalizations of some definitions such: j-ω-closure converge to a point,  j-ω-closure directed toward a set, almost  j-ω-converges to a set, almost  j-ω-cluster point, a set  j-ω-H-closed relative, j-ω-closure continuous mappings, j-ω-weakly continuous mappings, j-ω-compact mappings, j-ω-rigid a set, almost j-ω-closed mappings and  j-ω-perfect mappings. Also, we prove several results concerning it, where j ÃŽ{q, δ,a, pre, b, b}.

In modules there is a relation between supplemented and Ï€-projective semimodules. This relation was introduced, explained and investigated by many authors. This research will firstly introduce a concept of "supplement subsemimodule" analogues to the case in modules: a subsemimodule Y of a semimodule W is said to be supplement of  a subsemimodule X if it is minimal with the property X+Y=W. A subsemimodule Y is called a supplement subsemimodule if it is a supplement of some subsemimodule of W. Then, the concept of supplemented semimodule will be defined as follows: an S-semimodule W is said to be supplemented if every subsemimodule of W is a supplemen