and Institute for Mathematical Research (INSPEM) Universiti Putra Malaysia MALAYSIA 2,3 Department of Mathematics

Transcript

1 International Journal of Pure and Applied Mathematics Volume 82 No , ISSN: (printed version); ISSN: (on-line version) url: doi: PAijpam.eu PRIME GAMMA-NEAR-RINGS WITH (σ, τ)-derivations Isamiddin S. Rakhimov 1, Kalyan Kumar Dey 2, Akhil Chandra Paul 3 1 Department of Mathematics, FS and Institute for Mathematical Research (INSPEM) Universiti Putra Malaysia MALAYSIA 2,3 Department of Mathematics Rajshahi University Rajshahi-6205, BANGLADESH Abstract: Let N be a 2 torsion free prime Γ-near-ring with center Z(N) and let d be a nontrivial derivation on N such that d(n) Z(N). Then we prove that N is commutative. Also we prove that if d be a nonzero (σ,τ)-derivation on N such that d(n) commutes with an element aofn then ether d is trivial or a is in Z(N). Finally if d 1 be a nonzero (σ,τ)-derivation and d 2 be a nonzero derivation on N such that d 1 τ = τd 1, d 1 σ = σd 1, d 2 τ = τd 2, d 2 σ = σd 2 with d 1 (N)Γσ(d 2 (N)) = τ(d 2 (N))Γd 1 (N) then N is a commutative Γ-ring. AMS Subject Classification: 16Y30, 19W25, 16U80 Key Words: ring, prime ring, derivation 1. Introduction Bell and Mason [3] initiated the study of derivations in near-rings. The derivation inγ-near-rings was firstintroduced by Choand Jun[6]. Theystudied some Received: May 24, 2012 Correspondence author c 2013 Academic Publications, Ltd. url:

2 670 I.S. Rakhimov, K.K. Dey, A.C. Paul basic properties of prime Γ-near-rings. Ozturk and Jun in [14] discussed prime Γ-near-rings with derivations. They obtained commutativity properties on prime Γ-near-rings. Later Asci [1] studied Γ-near-rings with (σ,τ)-derivations and obtained some commutativity results. The commutativity properties of prime Γ-near-rings with derivations also have been investigated by Uckun, Ozturk and Jun [18]. Golbasi [10] obtained analogue of Posner s Theorem from [17]. The commutativity conditions of prime Γ- near-rings with generalized derivations have been studied in [7] (also see [8]). The paper [9] has dealt with generalized derivations in semiprime gamma-rings. In this paper we consider the prime Γ-near-rings with derivations to subject of commutativity conditions. The characterizations of prime Γ-near-rings with composition of two derivations are obtained. We also investigate conditions for prime Γ-near-ring to be commutative. Organization of the paper is as follows. Sections 2 and 3 contain brief review and definitions which will be used further. Some of these results have been presented by others researchers (for example, see [1]). Main results are in Sections 3 and 4. The results of Section 3 concern derivations in Γ-rings (Theorem 3.6), whereas Section 4 deals with (σ,τ)-derivations in prime Γ-nearrings (Theorems ). 2. Preliminaries A Γ-near-ring is a triple (N, +, Γ), where: (i) (N, +) is a group (not necessarily abelian), (ii) Γ is a non-empty set of binary operations on N such that for each α Γ, (N, +, α) is a left near-ring. (iii) aα(bβc) = (aαb)βc, for all a, b, c N and α, β Γ. Throughout this paper, N stands for a zero-symmetric left Γ-near-ring with multiplicative center Z(N). Recall that a Γ-near-ring N is prime if aγnγb = { 0 } implies that a = 0 or b = 0. Let σ and τ be two Γ-near-ring automorphisms of N. For a, b N and α Γ, [a, b] α, [a, b] α (σ,τ) and (a, b) will denote the commutator aαb bαa, aασ(b) τ(b)αa and a + b a b respectively. An additive mapping d: N N is called a derivation if d(aαb) = aαd(b) + d(a)αb holds for all a, b N, α Γ. An additivemappingd: N N iscalleda(σ, τ)-derivationifd(aαb) = σ(a)αd(b)+ d(a)ατ(b) holds for all a, b N, α Γ. In particular [a, b] (1,1) α = [a, b] α.

3 PRIME GAMMA-NEAR-RINGS WITH (σ, τ)-derivations Properties of Γ Rings with Derivations Lemma 3.1. Let d be an arbitrary additive endomorphism of N. Then d(aαb) = aαd(b)+d(a)αb, for all a, b N, α Γ if and only if d(aαb) = d(a)αb+aαd(b), for all a, b N, α Γ. Proof. Suppose d(aαb) = aαd(b)+d(a)αb for all a, b N, α Γ. Since aα(b+b) = aαb+aαb, d(aα(b+b)) = aαd(b+b)+d(a)α(b+b) = aαd(b)+aαd(b)+d(a)αb+d(a)αb and d(aαb+aαb) = d(aαb)+d(aαb) = aαd(b)+d(a)αb+aαd(b)+d(a)αb. Then, we obtain aαd(b)+d(a)αb = d(a)αb+aαd(b), so d(aαb) = d(a)αb+aαd(b), for all a, b N, α Γ. The converse part is proved similarly. Lemma 3.2. Suppose that d is a derivation on N. Then N satisfies the following right distributive laws: (i) (aαd(b) + d(a)αb)βc = aαd(b)βc + d(a)αbβc, (ii) (d(a)αb +aαd(b))βc = d(a)αbβc +aαd(b)βc for every a, b, c N and α, β Γ. Proof. (i) Consider d((aαb)βc) = aαbβd(c) + d(aαb)βc = aαbβd(c) +(aαd(b) + d(a)αb)βc, for all a, b, c N and α, β Γ, and d(aα(bβc)) = aαd(bβc) + d(a)αbβc = aα(bβd(c) + d(b)βc) + d(a)αb)βc for all a, b, c N and α, β Γ. = aαbβd(c) + aαd(b)βc + d(a)αbβc

4 672 I.S. Rakhimov, K.K. Dey, A.C. Paul From the above two relations we obtain for all a, b, c N and α, β Γ. (aαd(b) + d(a)αb)βc = aαd(b)βc + d(a)αbβc, (ii) Due to Lemma 3.1 we have, d((aαb)βc) = d(aαb)βc + aαbβd(c) = (d(a)αb + aαd(b))βc + aαbβd(c), for a, b, c N,α,β Γ, and d(aα(bβc)) = d(a)αbβc + aαd(bβc) = d(a)αbβc + aα(d(b)βc + bαd(c)) for every a, b, c N and α, β Γ. These two relations imply that for all a, b, c N and α, β Γ. = d(a)αbβc + aαd(b)βc + aαbβd(c), (d(a)αb + aαd(b)βc = d(a)αbβc + aαd(b)βc, Lemma 3.3. Let d be a derivation on a Γ-near-ring N and let u be not a left zero divisor in N. If [u, d(u)] α = 0 for every α Γ, then (a, u) is a constant for every a N. and Proof. From uα(u+a) = uαu+uαa, for all x N, α Γ, we get d(uα(u+a)) =uαd(u+a)+d(u)α(u+a) =uαd(u)+uαd(a)+d(u)αu+d(u)αa =uαd(u)+uαd(a)+uαd(u)+aαd(u) d(uαu+uαa) = d(uαu)+d(uαa) = uαd(u)+d(u)αu+uαd(a)+d(u)αa which gives uαd(a)+d(u)αu d(u)αu uαd(a) = 0, for all α Γ. Since d(u)αu = uαd(u), α Γ. This equation can be written as uα(d(a)+d(u) d(a) d(u)) = uαd(a+u a u) = uαd((a,u)) = 0.

5 PRIME GAMMA-NEAR-RINGS WITH (σ, τ)-derivations 673 Thus d((a, u)) = 0. Hence (a, u) is constant. This completes the proof. Lemma 3.4. Let d be a (σ, τ)-derivation on a N and a N. Then (τ(a)αd(b) + d(a)ασ(b))βσ(c) = τ(a)αd(b)βσ(c) + d(a)ασ(b)βσ(c), for a, b, c N,α, β Γ. Proof. Consider d((aαb)βc) = τ(aαb)βd(c) + d(aαb)βσ(c) for all a, b, c N, α, β Γ, and d(aα(bβc)) = τ(a)αd(bβc) + d(a)ασ(bβc) for all a, b, c N, α, β Γ. Since d((aαb)βc) = d(aα(bβc)), we have = τ(a)ατ(b)βd(c) +(τ(a)αd(b) + d(a)ασ(b))βσ(c), = τ(a)ατ(b)βd(c) + τ(a)αd(b)βσ(c) + d(a)ασ(b)βσ(a) (τ(a)αd(b) + d(a)ασ(b))βσ(c) = τ(a)αd(b)βσ(c) + d(a)ασ(b)βσ(c), for all a,b, c N, α, β Γ. This completes the proof. Lemma 3.5. Suppose that N is a prime Γ-near-ring. (i) any nonzero element of the center of N is not zero divisor. (ii) If there exist a nonzero element of Z(N) such that a+a Z(N), then (N, +) is commutative. (iii) If d 0 be a derivation on N. Then aγd(n) = {0} implies a = 0, and d(n)γa = {0} implies a = 0. (iv) Let N be 2-torsion free and d be a derivation on N with d 2 = 0. Then d = 0. Proof. (i) If a Z(N) { 0 } and aαb = 0, b N, α Γ, then aαcβb = 0, b, c N, α Γ. Therefore we have, aγnγb = 0. Since N is prime and a 0 then b = 0. (ii) Let a Z(N) { 0 } be an element such that a+a Z(N) and let b, c N, α Γ. By the distributivity of N we get (b+c)α(a +a) = bα(a+a)+ cα(a+a) = bαa+bαa+cαa+cαa = aαb+aαb+aαc+aαc = aα(b+ b+c+c).

6 674 I.S. Rakhimov, K.K. Dey, A.C. Paul Next, since a+a Z(N) then (b+c)α(a+a) = (b+c)αa+(b+c)αa = aα(b+c)+aα(b+c) = aα(b+c+b+c). Taking into account the last two we have, b+b+c+c = b+c+b+c and so b+c = c+b. Hence (N, +) is commutative. (iii) Let aγd(n) = { 0 }, b, c N and α, β Γ. Then 0 = aαd(bβc) = aαbβd(c) + aαd(b)βc = aαbβd(c). Thus aγnγd(n) = { 0 }, by using the primeness of N and d(n) 0 we get a = 0. The part (iii) can be proved similarly. (iv) Note that for any a, b N, α Γ, 0 = d 2 (aαb) = d(aαd(b)+d(a)αb) = aαd 2 (b)+d(a)αd(b)+d(a)αd(b)+ d 2 (a)αb. Therefore we have 2d(a)αd(b) = 0, for all a, b N, α Γ. Since N is 2-torsion free we get d(a)αd(b) = 0, for each a N, α Γ. By using (iii) we obtain d = 0. Theorem 3.6. Let N be a prime Γ-near-ring and let d be a nontrivial derivations on N such that d(n) Z(N). Then (N, +) is commutative. Moreover if N is 2-torsion free, then Nis a commutative Γ-ring. Proof. Let a be a non-constant and c be a constant elements of N. Then d(aαc) = aαd(c) + d(a)αc = d(a)αc Z(N), α Γ. Since d(a) Z(N) { 0 }, it follows easily that c Z(N). Note that if c is a constant so c+c is a constant. It follows from lemma 3.5(ii) that (N, +) is commutative, provided that there exists a nonzero constant. Let us provide the last. Suppose, that 0 is the only constant. Since d is obviously commuting, it follows from the lemma 3.3 that all u which are not zero divisors belong to by Z(N). In particular, if a 0, d(a) Z(N). But then for all b N, d(b)+d(a) d(b) d(a) = d((b, a)) = 0, hence (b, a) = 0. Now, we assume that N is 2-torsion free. By Lemma 3.1, (aαd(b) + d(a)αb)βc = aαd(b)βc + d(a)αbβc for all a, b, c N, α, β Γ, and using the fact that d(aαb) Z, α Γ, we get cαaβd(b) + cαd(a)βb = aαd(b)βc + aαd(b)βc, α, β Γ. Since (N, +) is commutative and d(n) Z(N), we obtain d(b)α[c, a] β = d(a)α[b, c] β for all a, b, c N, α, β Γ. Suppose now that N is not commutative. Choosing b, c N, with [b, c] β 0, β Γ, and letting u = d(a), we get d 2 (a)α[b, c] β = 0, for all a N, α, β Γ, and since the central elements d 2 (a) cannot be a nonzero divisor of zero, we conclude that d 2 (a) = 0 for all a N. By lemma 3.5(iv), this is not true for nontrivial d.

7 PRIME GAMMA-NEAR-RINGS WITH (σ, τ)-derivations Γ-Near-Rings with (σ, τ)-derivations Lemma 4.1. Suppose that N is a prime Γ-near-ring with a nonzero (σ,τ)- derivation d and a N. If at least one of the following (i) d(n)γσ(a) or (ii) aγd(n) is zero then a = 0. Proof. (i) Consider 0 =d(bβc)ασ(a) = (τ(b)βd(c) + d(b)βσ(c))ασ(a) =τ(b)βd(c)ασ(a) + d(b)βσ(c)ασ(a), for all a, b, c N, α, β Γ. Since d(c)ασ(a) = 0, we have d(b)βσ(c)ασ(a) = 0, i.e., d(b)γnγσ(a) = 0. (Bearing in mind that σ is an automorphism). Since d 0 and N is prime, we get σ(a) = 0 which gives a = 0. The proof of (ii) is similar. Theorem 4.2. Suppose that d is a nonzero (σ, τ)-derivation of a prime Γ-near-ring N and a N, α Γ. Let [d(n), a] (σ,τ) α = 0, then d(a) = 0 or a Z(N). Proof. Note that for all b N and α Γ, aαb N. Therefore, from [d(n), a] (σ,τ) α = 0 we have, d(aαb)βσ(a) = τ(a)βd(aαb) for β Γ. Hence that is (τ(a)αd(b) + d(a)ασ(b))βσ(a) = τ(a)β(τ(a)αd(b) + d(a)ασ(b)). Then by Lemma 3.4, we have τ(a)αd(b)βσ(a) + d(a)ασ(b)βσ(a) = τ(a)βτ(a)αd(b) + τ(a)βd(a)ασ(b). Due to the condition of the theorem, we have τ(a)ατ(a)βd(b) + d(a)ασ(b)βσ(a) = τ(a)βτ(a)αd(b) + d(a)ασ(a)βσ(b), d(a)ασ([b,a] β ) = 0 (1) for all b N, α, β Γ. Substituting bδc, (c N, δ Γ) for a in (1), we have d(a)ασ(b)βσ([c, a] α ) = 0 for all b, c N, α, β, δ Γ. Since σ is automorphism we get d(a) = 0 or a Z(N). This completes the proof.

8 676 I.S. Rakhimov, K.K. Dey, A.C. Paul Lemma 4.3. Suppose that N is a prime Γ-near-ring and d 1 is a nonzero (σ, τ)-derivation and d 2 is a derivation of N such that d 1 (a)γσ(d 2 (b))+τ(d 2 (a))γd 1 (b) = 0 for all a, b N. Then (N, +) is commutative. Proof. Consider d 1 (a)γσ(d 2 (b)) +τ(d 2 (a))γd 1 (b) = 0 for all α Γ. Replacing b by u+v we have, 0 = d 1 (a)ασ(d 2 (u+v))+τ(d 2 (a))αd 1 (u+v) = d 1 (a)ασ(d 2 (u))+d 1 (a)ασ(d 2 (v))+τ(d 2 (a))αd 1 (u)+τ(d 2 (a))αd 1 (v), for u, v N, α Γ. Using the condition of the lemma, we get 0 = d 1 (a)ασ(d 2 (u))+d 1 (a)ασ(d 2 (v)) d 1 (a)ασ(d 2 (u)) d 1 (a)ασ(d 2 (v)). As result we get, d 1 (a)ασ(d 2 (u,v)) = 0 (2) for all a, u, v N, α Γ. By Lemma 4.1(i), we obtain that d 2 (u, v) = 0 for all u, v N. Note that for any w N, we have d 2 (wαu, wαv) = 0. Using (2), we obtain that d 2 (wα(u, v)) = 0. This yields d 2 (w)α(u, v) = 0, for all w, u, v N, α Γ. By Lemma 3.5(iii), we get (u, v) = 0, for all u, v N. Theorem 4.4. Let N be a prime Γ-near-ring with 2N { 0 }, d 1 be a (σ, τ)-derivation and d 2 be a derivation such that d 1 τ = τd 1, d 1 σ = σd 1, d 2 τ = τd 2 and d 2 σ = σd 2. If d 1 (a)γσ(d 2 (b)) +τ(d 2 (a))γd 1 (b) = 0 for all a, b N, then either d 1 = 0 or d 2 = 0. Proof. Let d 1 0 and d 2 0. According to Lemma 4.3 we know that (N, +) is commutative. Now, for all u, v N we take a = uβv in d 1 (a)γσ(d 2 (b)) +τ(d 2 (a))γd 1 (b) = 0 to obtain 0 = d 1 (uβv)ασ(d 2 (b))+τ(d 2 (uβv))αd 1 (b) = (τ(u)βd 1 (v)+d 1 (u)βσ(v))ασ(d 2 (b))+τ(uβd 2 (v)+d 2 (u)βv)αd 1 (b). Using the left distributive law, we get, 0 = τ(u)βd 1 (v)ασ(d 2 (b))+d 1 (u)βσ(v)ασ(d 2 (b))+τ(u)βτ(d 2 (v))ασd 1 (b)

9 PRIME GAMMA-NEAR-RINGS WITH (σ, τ)-derivations 677 for all u, v, b N, α, β Γ. Hence, +τ(d 2 (u))βτ(v)αd 1 (b), 0 = τ(u)β(d 1 (v)ασ(d 2 (b)+τ(d 2 (v))αd 1 (b))+d 1 (u)βσ(v)ασ(d 2 (b)) Now due to the condition of the theorem, we obtain +d 2 (τ(u))βτ(v)αd 1 (b). d 1 (u)βσ(v)ασ(d 2 (b))+d 2 (τ(u))βτ(v)αd 1 (b) = 0, (3) for all b, u, v N, α, β Γ. Replacing b by bδt, t N, δ Γ, in (3) we have 0 =d 1 (u)αβσ(v)ασ(d 2 (bδt))+d 2 (τ(u))βτ(v)αd 1 (bδt) =d 1 (u)βσ(v)ασ(b)δσ(d 2 (t))+d 1 (u)βσ(v)ασ(d 2 (b))δσ(t) +d 2 (τ(u))βτ(v)ατ(y)δd 1 (t)+d 2 (τ(u))βτ(v)αd 1 (b)δσ(t) =d 1 (u)βσ(vβb)δσ(d 2 (t))+d 2 (τ(u))βτ(vβb)δd 1 (t) +d 1 (u)βσ(v)ασ(d 2 (b))δσ(t) +d 2 (τ(u))βτ(v)βd 1 (b)δτ(t). Using (3) we again obtain, d 1 (u)βσ(v)βσ(d 2 (b))βσ(t)+d 2 (τ(u))βτ(v)αd 1 (b)δσ(t) = 0, (4) for all u, v, b, t N, α, β, δ Γ. Substituting d 1 (t) for t in (4), we get d 1 (u)ασ(v)βσ(d 2 (b))δσ(d 1 (t))+d 2 (τ(u))βτ(v)d 1 (b)δσ(d 1 (t)) = 0 (5) for all u, v, b, t N, α, β, δ Γ. Now if we take τ(b) instead of y in (5), we have, d 1 (u)ασ(v)βσ(d 2 (τ(b)))δσ(d 1 (t))+d 2 (τ(u))βτ(v)αd 1 (τ(b))δσ(d 1 (t)) = 0, (6) for all u, v, b, t N, α Γ. Substituting v and b in (3) by vγd 1 (b) and σ(t) respectively, we obtain, d 1 (u)ασ(v)βσ(d 1 (y))δσ(d 2 (σ(t)))+d 2 (τ(u))βτ(v)δτ(d 1 (y))γd 1 (σ(t)) = 0, (7) for all u, v, b, t N, α, β Γ.

10 678 I.S. Rakhimov, K.K. Dey, A.C. Paul Now, if we subtract (6) from (7) and make use d 1 σ = σd 1, d 1 τ = τd 1, then Therefore, d 1 (u)ασ(v)δ(d 1 (σ(y))βσ(d 2 (σ(t)) σ(d 2 (τ(y))βσ(d 1 (t)) = 0. d 1 (N)ΓNΓ(σ(d 1 (b)γd 2 (σ(t)) d 2 (τ(b))βd 1 (t)) = 0, (8) for all t, b N. Since N is a prime Γ-near-ring and d 1 0 we conclude that, d 1 (b)αd 2 (σ(t)) d 2 (τ(b))βd 1 (t) = 0. Using d 2 σ = σd 2 and the hypothesis, we get d 1 (b)βσ(d 2 (t)+d 2 (t)) = 0, (9) for all b, t N. Due to Lemma 4.1(i) and d 1 0 we get, d 2 (t) +d 2 (t) = 0, for all t N. Hence 0 = d 2 (sαt) + d 2 (sαt) = d 2 (s)α(t + t) for all s, t N, α Γ, and so d 2 (N)α(t+t) = 0, for all t N. If we apply Lemma 3.5(iii), we get t+t = 0 for all t N, which contradicts that 2N {0}. This completes the proof. Theorem 4.5. Let N be a prime Γ-near-ring with 2N { 0 }, d 1 be a (σ, τ)-derivation and d 2 beaderivation such that d 1 τ = τd 1, d 1 σ = σd 1, d 2 τ = τd 2 and d 2 σ = σd 2.(i) If d 1 (a)γσ(d 2 (b)) +τ(d 2 (a))γd 1 (b) = 0 for all a, b N, then either d 1 = 0 or d 2 = 0.(ii) If d 1 d 2 acts as a (σ, τ)-derivation on N, then either d 1 = 0 or d 2 = 0. and Proof. Consider d 2 d 1 (aαb) = τ(a)αd 2 d 1 (b)+d 2 d 1 (a)ασ(b), d 2 d 1 (aαb) = d 2 (τ(a))αd 1 (b)+τ(a)αd 2 d 1 (b)+d 2 d 1 (a)ασ(b)+d 1 (a)αd 2 (σ(b)). These two expressions give τ(d 2 (a))αd 1 (b) +d 1 (a)ασ(d 2 (b)) = 0, for all a, b N. Now due to Theorem 4.4, we get d 1 = 0 or d 2 = 0. Theorem 4.6. Let N be a prime Γ-near-ring with a nonzero (σ, τ)- derivation d such that d(aαb) = d(bαa) for all a,b N, α Γ, then N is a commutative Γ-near-ring.

11 PRIME GAMMA-NEAR-RINGS WITH (σ, τ)-derivations 679 Proof. Let us consider τ(a)αd(b) + d(a)ασ(b) = τ(b)αd(a) + d(b)ασ(a), (10) for all a,b N, α Γ. We substitute bβa for a in (10) and obtain τ(b)βτ(a)αd(b) + d(bβa)ασ(b) = τ(b)βd(bαa) + d(b)ατ(b)ασ(a). By the partial distributive law for a,b N, α, β Γ and using d(bαa) = d(aαb) we get τ(b)βτ(a)αd(b) + τ(b)βd(a)ασ(b) + d(b)ασ(a)βσ(b) Therefore, = τ(b)βτ(a)αd(b) + τ(b)βd(a)ασ(b) + d(b)ασ(b)βσ(a). d(b)ασ(a)βσ(b) = d(b)ασ(b)βσ(a), (11) for all a,b N, α, β Γ. Replacing a by aδc, c N in the last equality we get d(b)βσ(a)βσ(c)βσ(b) = d(b)ασ(b)ασ(a)ασ(c) = d(b)ασ(a)βσ(b)ασ(c) and so d(b)γnγσ([c, b] α ) = 0, for all a, c N, α, β Γ. By the primeness of N, we obtain d(b) = 0 or b Z(N). Since d is a nonzero (σ, τ)-derivation on N, we have b Z(N). So, N is a commutative Γ-near-ring. Theorem 4.7. Let N be a prime 2-torsion free Γ-near-ring and d 1 be a nonzero (σ, τ)-derivation, d 2 a nonzero derivation on N such that d 1 τ = τd 1,d 1 σ = σd 1, d 2 τ = τd 2 and d 2 σ = σd 2. If d 1 (a)γσ(d 2 (b)) = τ(d 2 (a))γd 1 (b) for all a,b N, then N is a commutative Γ-ring. Proof. Consider [d 1 (a), d 2 (b)] (σ,τ) α = 0, for all a,b N, α Γ. Applying Theorem 3.6, we obtain that d 1 d 2 (N) = 0 or d 2 (N) Z(N). Since d 1 and d 2 are nonzero derivations, we have d 2 (N) Z(N). Now according to Theorem 4.2 we get the commutativity of N. Acknowledgments The second named author thanks the Institute for Mathematical Research (IN- SPEM), UPM, Malaysia for the hospitality during that this paper has been written. The research was supported by FRGS grant FR MOHE, Malaysia.

12 680 I.S. Rakhimov, K.K. Dey, A.C. Paul References [1] M. Asci, Γ (σ, τ)-derivation on gamma near ring, International Math. Forum, 2, No. 3 (2007), [2] J. Bergen, Derivations in prime rings, Canad. Math. Bull., 26, No. 3(1983), [3] H.E. Bell, G. Mason, On derivations in near-rings, near-rings and nearfields, (Ed. Gerhard Betsch), North-Holland, Amsterdam (1987), [4] Y.U. Cho, A study on derivations in near-rings, Pusan Kyongnam Math. J., 12, No. 1 (1996), [5] Y.U. Cho, Some conditions on derivations in prime near rings, J. Korea Soc. Math. Educ., Ser. B, Pure Appl. Math., 8, No. 2 (2001), [6] Y.U. Cho, Y.B. Jun, Gamma derivation in prime and semiprime gamma near-rings, Indian J. Pure and Appl. Math., 33, No. 10 (2002), [7] K.K. Dey, A.C. Paul, I.S. Rakhimov, On commutativity in gamma-nearring with symmetric bi-(σ, τ)-derivations, Far East J. of Math. Sciences, 65, No. 2, [8] K.K. Dey, A.C. Paul, I.S. Rakhimov, On prime gamma-near-rings with generalized derivations, International Journal of Math. and Math. Sci. (2012), doi: /2012/ [9] K.K. Dey, A.C. Paul, I.S. Rakhimov, Generalized derivations in semiprime gamma-rings, International Journal of Math. and Math. Sci. (2012), doi: /2012/ [10] O. Golbasi, On a theorem of Posner for 3-prime near rings with (σ,τ)- derivation, Hacettepe J. of Mathematics and Statistics, 36, No. 1 (2007), [11] O. Golbasi, N. Aydin, Results on prime near-ring with (σ, τ)-derivation, Math. J. Okayama Univ., 46 (2004), 1-7. [12] Y.B. Jun, K.H. Kim, Y.U. Cho, On gamma-derivation in gamma-nearrings, Soochow Journal of Mathematics, 29, No. 3 (2003), [13] M. Kazaz, A. Alkan, Two sided Γ-α-derivations in prime and semiprime Γ-near-rings, Commun. Korean Math. Soc., 23, No. 4 (2008),

13 PRIME GAMMA-NEAR-RINGS WITH (σ, τ)-derivations 681 [14] M.A. Ozturk, Y.B. Jun, On the trace of symmetric bi-derivations in nearrings, Inter. J. Pure and Appl. Math., 17, No. 1 (2004), [15] M.A. Ozturk, Y.B. Jun, On generalized symmetric bi-derivations in primerings, East Asian Math. J., 15, No. 2 (1999), [16] G. Pilz, Near-Rings, North-Holland Mathematics Studies, 23, North- Holland, Amsterdam (1983). [17] E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), [18] M. Uckun, M.A. Ozturk, Y.B. Jun, On prime gamma-near-rings with derivations, Commun. Korean Math. Soc., 19, No. 3 (2004),

14 682