γ-ORTHOGONAL FOR K-DERIVATIONS AND K-REVERSE DERIVATIONS

In this paper, we introduce definitions of γ-orthogonality for two pairs of k-derivations, generalized k-derivations, and k-reverse derivations. And we present some results concerning with these notions on γ-semiprime gamma ring.


Introduction
Let M and Γ be two additive abelain groups. M is said to be a Γ-ring in the sense of Barnes [1] if there exists a mapping of M × Γ × M → M satisfying these two conditions for all a , b , c ∊ M, α , β ∊ Γ: (1) ( a + b)αc = aߙc + bߙc a(ߙ + ߚ )b = aߙb + aߚb aߙ (b + c) = aߙb + aߙc (2) (aߙb)ߚc = aߙ(bߚc) In addition, if there exists a mapping of Γ × M × Γ → Γ such that the following axioms hold for all a , b , c ∊M , α , β ∊ Γ: (3) (aߙb)ߚc = a(ߙbߚ )c (4) aߙb = 0 for all a, b ∊ M implies ߙ = 0 where ߙ ∊ Γ.
Then M is called a Γ-ring in the sense of Nobusawa [2]. If a Γ-ring M in the sense of Barnes satisfies only the condition (3), then it is called weak Nobusawa Γ-ring [3]. We assume that all gamma rings in this paper are weak Nobusawa Γ-ring unless otherwise specified.
Let M be Γ-ring, M is said to be a Γ-prime gamma ring if a ΓM Γb = 0 with a, b ∊ M implies that either a = 0 or b = 0 [4], and M is called a Γ-semiprime gamma ring if a ΓM Γa = 0 with a ∊ M implies a = 0 [4]. A weak Nobusawa Γ-ring M is said to be a γ-prime gamma ring if there exists a non-zero element γ in Γ such that a γM γ b = 0 with a, b ∊M implies that either a = 0 or b = 0 [5] and is called a γsemiprime gamma ring if there exists a non-zero element γ in Γ such that a γM γ a = 0 with a ∊M implies that a = 0 . And a Γ-ring M is said to be a 2-torsion free if 2a = 0, a ∊M implies a = 0.
Recall that from [6], an additive mapping d : M → M is called a derivation on M if d(aαb) = d(a)ߙb + aߙd(b) for all a, b ∊M, α ∈ Γ and a reverse derivation on M if d(aαb) = d(b)ߙ a + bߙ d(a) for all a, b ∊M, α ∈ Γ [7]. Also an additive mapping D: M→M is said to be a generalized derivation if there exists a derivation d on M such that D(aαb) = D(a)ߙb + aߙd(b) for all a, b ∊M, α ∈ Γ [8]. In 2000, Kandamar [9] firstly introduced the notion of a k-derivation for a gamma ring in the sense of Barnes [10] introduced the notion of generalized k-derivations for gamma rings. Also in [11] presented the notion of a k-reverse derivation for a gamma ring.
In this work, we define γ-orthogonality for two pairs k-derivations, generalized k-derivations and kreverse derivations for a weak Nobusawa gamma ring. And we obtain some results on 2-tortion free γsemiprime Γ-ring.

γ-Orthogonal generalized k-derivations
Now we introduce the notion of γ-orthogonal k-derivations as follows.
In the following theorem we give characterization of γ-orthogonal generalized k-derivations on Γring.
(ii) Similar way used in the proof of (i).