International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN(P): 2249-6955; ISSN(E): 2249-8060 Vol. 4, Issue 1, Feb 2014, 43-50 © TJPRC Pvt. Ltd. ON SEMIPRIME GAMMA NEAR- RINGS MEHSIN JABEL ATTEYA 1 , ATHEER GHAZI HUSSEIN 2 & DALAL IBRAHEEM RASEN 3 1,3 Department of Mathematics, AL-Mustansiriyah University, College of Education, Iraq 2 Department of Mathematics, Waset University, College of Education, Iraq ABSTRACT The main purpose of this paper is to study and investigate some results concerning permuting tri-derivations on semi prime - near-rings M, when M be a 3-torsion free semi prime -ring with satisfying xyz=xyz for all x,y, zM, ,.If there exists a permuting tri-derivation D:M×MM→M, where d is the trace of D KEYWORDS: Tri-Derivation, Semi Prime -Ring, Commutative Ideal, Commuting Map, Permuting Map 2000 Mathematics Subject Classifications: 16W20, 16W25 1. INTRODUCTION Every ring is a right near-ring (resp. a left near-ring). But in general the converse is not true. A right near-ring (resp. a left near-ring) not be a ring. In [2] Bell and Mason introduced the notion of derivations in near-rings. They obtained some basic properties of derivations in near-rings. Then Mustafa [11] investigated some commutativity conditions for a G-near-ring with derivations. Cho [5] studied some characterizations of G-near-rings and some regularity conditions. In classical ring theory, Posner [9], Herstein [6], Bergen [4], Bell and Daif [1] studied derivations in prime and semi prime rings and obtained some commutativity properties of prime rings with derivations. In near ring theory, Bell and Mason [2], and also Cho [5] worked on derivations in prime and semi prime near-rings. Gamma rings were first introduced by Nabusawa [12] and then Barnes [13] generalized the definition of -rings. The ideal and other definitions of the concepts in -rings we refer to [13].Eduard Domi[14]proved ,that if ideal I of -near-ring M is maximal, then it is prime or M  M = I. Kalyan Kumar Dey and Akhil Chandra Paul [15]proved, let N be a semi prime -near-ring and let U be a nonzero N-subset of N. If a be an element of N(U) such that Uaa = {0} (or aaU = {0}), where N(U) is the normalizer of U, then a = 0. Yong Uk Cho[16 ]proved, Every left ideal of a P(1,2) -near-ring is an ideal,where M is said to be a P(r;m) -near-ring if there exist positive integers r, m such that xrM = Mxm for all x M. Kalyan Kumar Dey[17]proved ,let M be a 2-torsion-free semi prime Γ-ring and D : M → M be an additive mapping which satisfies D(xαx)= D(x)αx for all x ∈M, α ∈ Γ. Then D is a left centralizer. Young Bae Jun , Kyung Hokim and Yong Uk Cho[18]proved , if d is a Γ-derivation on M, then d(xγy) = d(x)γy + xγd(y), for all x, y ∈ M and γ ∈ Γ, where M is Γ-near- ring. As a generalization of near-rings, Γ-near-rings were introduced by Satyanarayana [23]. Booth (together with Groenewald) have studied several aspects in Γ-near-rings (see [19, 20, 21, 22]). Ozturk, Sapanci, Soyturk and Kim [24] studied on symmetric bi-derivations on prime -rings. Some fruitful results of prime -rings were obtained by them. Ozturk [25] obtained some properties concerning to the mapping permuting tri-derivations on prime and semi prime -rings. Permuting tri-derivations in prime and semiprime -rings had been studied by Sapanci, M.A. Ozturk and Y.B. Jun [26]. Some remarkable results of these -rings were obtained by them. In this paper is to study and investigate some results concerning permuting tri-derivations on semiprime - near-rings and prime - near-rings, we give some results about that.