Lemma 2.3 . Let M be a 2- torsion free **Γ**- ring satisfying the condition (*) and let U be a **Lie** ideal of M such that uαu ∈ U for all u ∈ U and α ∈ **Γ**. Let d : M → M be a **Jordan** **k**-derivation on U of M . Then for all u, v, w ∈ U and α, β ∈ **Γ** , we have the following : (i)d(uαv +vαu) = d(u)αv +uk(α)v +uαd(v) +d(v)αu + vk(α)u + vαd(u)

Abstract. In this paper, we analyzed the basic properties and related theorem of **Lie** **ideals** on **prime** **Γ** -**rings** with **Jordan** right **derivations**. We mainly focused on the characterizations of 2-torsion free **prime** **Γ** -**rings** by using **Lie** **ideals** and **Jordan** right **derivations**. Our main objective is to prove the theorem that if ܯ be a 2 -torsion free **prime** **Γ** -ring and ܷ be a **Lie** ideal of ܯ such that ݑߙݑ ∈ ܷ for all ݑ, ݒ ∈ ܷ and ߙ ∈ **Γ** and ݀: ܯ → ܯ is an additive mapping such that ݀(ݑߙݑ) = 2 ݀(ݑ)ߙݑ for all ݑ ∈ ܷ and ߙ ∈ **Γ** , then ݀(ݑߙݒ) = ݀(ݑ)ߙݒ + ݀(ݒ)ߙݑ for all ݑ, ݒ ∈ ܷ and ߙ ∈ **Γ** .

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1. Introduction. Throughout the present paper, R will denote an associative ring with centre Z(R). We will write for all x, y ∈ R, [x, y] = xy − yx and x ◦ y = xy + yx for the **Lie** product and **Jordan** product, respectively. A ring R is said to be **prime** if aRb = (0) implies that a = 0 or b = 0. A ring R is said to be 2-torsion-free if whenever 2a = 0, with a ∈ R, then a = 0. An additive subgroup J of R is said to be a **Jordan** ideal of R if u ◦ r ∈ J, for all u ∈ J and r ∈ R. An additive mapping d : R → R is called a derivation (resp., **Jordan** derivation) if d(xy) = d(x)y + xd(y) (resp., d(x 2 ) = d(x)x + xd(x)) holds for all x, y ∈ R. Let θ, φ be endomorphisms of

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In [4], Bell and Kappe proved that if d is a derivation of a semiprime ring R which is either an endomorphism or an anti-endomorphism on R, then d = 0; whereas, the behavior of d is somewhat restricted in case of **prime** **rings** in the way that if d is a derivation of a **prime** ring R acting as a homomorphism or an anti-homomorphism on a non-zero right ideal U of R, then d = 0. Asma et. al. [1] extended this result of **prime** **rings** on square closed **Lie** **ideals**. Afterwards, the said result was extended to σ-**prime** **rings** by Oukhtite et. al. in [11].

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A well known result of Posner [18] states that a **prime** ring admitting a nonzero centralizing derivation must be commutative. This theorem indicates that the global structure of a ring R is often tightly connected to the behaviour of additive mappings defined on R. Following this line of investigation, several authors studied **derivations** and generalized **derivations** acting on appropriate subsets of the ring.

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Firstly, E.C. Posner [17] proved pioneer results on derivation in **prime** **rings**. He established relation between derivation on a ring and structure of that ring. Many authors have generalized Posner’s theorems, for suitable subsets of R, as ideal, left ideal, **Lie** ideal and **Jordan** ideal, further information can be found in ([2], [5],[9],[12],[15],[16], [18],[20]). In [6] Bell and Kappe proved that if a deriva- tion d of a **prime** ring R which acts as homomorphisms or anti-homomorphisms on a nonzero right ideal of R then d = 0 on R. In [14] Nadeem-ur rehman generalized Bell and Kappe result by taking generalized derivation instead of derivation. Precisely he proved, let R be a 2-torsion free **prime** ring and I be a non zero ideal of R. Suppose F : R → R is a nonzero generalized derivation with non zero derivation d. If F acts as a homomorphism or anti-homomorphism on I then R is commutative. In this sequence, in 2001 Ashraf and Rehman [4], proved that if R is a **prime** ring with a non-zero ideal I of R and d is a derivation of R such that either d(xy) ±xy ∈ Z(R) for all x, y ∈ I or d(xy)±yx ∈ Z(R) for all x, y ∈ I, then R is commutative. Again, Asraf et al. [3] proved that if R is a **prime** ring which is 2 torsion free and F is a generalized derivation associated with derivation d on R. If F satisfies any one of the following conditions: (i) F(xy) − xy ∈ Z(R); (ii) F (xy) − yx ∈ Z(R); (iii) F (x)F (y) − xy ∈ Z(R); (iv) F(x)F(y) − yx ∈ Z(R), for all x, y ∈ I, where I is an ideal of R, then R is commutative. Recently, Albas [17] studied following identities in **prime** **rings**: (i) F(xy) ± F (x)F (y) ∈ Z(R); (ii) F (xy) ± F(y)F (x) ∈ Z (R), for all x, y ∈ I, a non zero ideal of R.

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In [3], Bergen et al. investigate the relationship between the **derivations** and **Lie** **ideals** of a **prime** ring, and obtain some useful results. In [4], P. H. Lee and T. **K**. Lee get six suﬃcient conditions of central **Lie** ideal, which extend some results of commutativity on a **prime** ring. Motivated by the above, in this paper, we extend M. Ashraf ’s results to a **Lie** ideal of a **prime** ring. Throughout this paper, R will be a **prime** ring and U will always denote a **Lie** ideal of R.

Where is an inner derivation. An additive mapping : → is called a generalized derivation associated with a derivation if ( ) = ( ) + ( ) , for all , ∈ . An additive mapping : → is called a left generalized derivation associated with a derivation if ( ) = ( ) + ( ) , for all , ∈ . It should be interesting to extend some results concerning these notions to left generalized **derivations**. Throughout the present paper we shall make extensive use of the following basic commutator identities without any specific mention:

Although it is possible to prove results about semi-**prime** **ideals** that are analogous to all of those established for **prime** **ideals** proceeding section, we shall present only a few that are essential for later applications. First, let us state the following theorem whose proof we omit since it can be established by very easy modifications of the proof of theorem 3.2.

such that [d(x), x] ∈ Z(R) for all x ∈ R, then R is commutative. An analogous result for centralizing automorphisms on **prime** **rings** was obtained by Mayne [28]. A number of authors have extended these theorems of Posner and Mayne; they have showed that **derivations**, automorphisms, and some related maps cannot be centralizing on certain subsets of noncommutative **prime** (and some other) **rings**. For these results we refer the reader to ([10], [12], [15], [25], where further references can be found). In [14], the description of all centralizing additive maps of a **prime** ring R of characteristic not 2 was given and subsequently in [4] the characterization for semiprime **rings** of characteristic not 2 was given. It was shown that every such map f is of the form f (x) = λx + µ(x), where λ ∈ C, the extended centroid of R, and µ is an additive map of R into C (see also [12] where similar results for some other **rings** are presented). Recently, some authors have obtained commutativity of **prime** and semiprime **rings** with **derivations** and generalized **derivations** satisfying certain polynomial identities (viz., [1], [2], [3], [5], [6], [7], [8], [9], [16], [17], [18], [19], [20], [24], [25], [27], [29], [31] and [32]). The main objective of the present paper is to investigate commutativity of **prime** and semiprime **rings** satisfying certain identities involving additive mappings and **derivations**.

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This note is motived by the previous cited results. Our main theorem gives a general- ization of Lanski’s result to the case when (dδ) is a **Lie** derivation of the subset [I,I ] into R, where I is a nonzero right ideal of R and the characteristic of R is di ﬀ erent from 2. The statement of our result is the following.

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One of the results required along the way is of independent interest. We prove (Lemma 3.1) that in a Noetherian ring an ideal with zero right annihilator and satisfying (a weak form of ) the right AR-property contains a right regular element. Our methods require Theorem A to be first proved for a maximal ideal. Extending the result to a general **prime** ideal presents a technical challenge. Since it is not yet known whether the cliques in a Noetherian PI-ring are localisable, a direct localisation approach is not available to us. We sidestep this difficulty by employing a trick of Goodearl and Stafford. This device guarantees that the **prime** ideal being examined extends to a **prime** ideal which belongs to a localisable clique in a polynomial extension of the given Noetherian PI-ring. With the authors’ permission an account of this method is included here.

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The set Z(R) = {a ∈R| aαb = bαa, ∀b ∈R, andα∈**Γ**} is called the center of R.A **Γ**-ring R is called **prime** if aΓRΓb = 0 with a, b ∈R implies a = 0 or b = 0, and R is called semi **prime** if aΓRΓa = 0 with a ∈R implies a = 0. The notion of a (resp. semi-) **prime** **Γ**-ring is an extension for the notion of a (resp. semi-) **prime** ring. In [1] F.J.Jing defined a derivation on **Γ**-ring as follows, an additive map d from a **Γ**- ring R into itself is called a derivation on R if d(aαb) = d(a)αb +aαd(b) , holds for all a,b∈R and α∈**Γ**, and in [2] S.

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Theorem.3.3. Let n ≥2 be any fixed positive integer. Let R be an n! – torsion free **prime** ring and I be any non zero two sided ideal of R. Suppose that there exist a non zero symmetric skew reverse n-derivation Ä: R n →R, associated with an antiautomorphism á * . Let ä denote the trace of such that ä is commuting on I and [ ä (x), á * (x)] Z(R), for all x I then R must be commutative.

The notion of a ring a concept more general than a ring was definedby Nobusawa [7]. As a generalization of near **rings**, **rings** were introduced by satyanarayana[8]. The derivation of a near ring has been introduced by Bell and Mason[2]. They obtained some basic properties of **derivations** in near ring. The recent literature contains numerous results on commutativity in **prime** and semiprime **rings** admitting suitably constrained **derivations** and generalized **derivations**, and several authors have proved comparable results on near-**rings**. Some of our results, which deal with conditions on **derivations**, extend earlier commutativity results involving similar conditions on **derivations** and semi-**derivations**.

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Recently, L.Oukhtite and S.Salhi [6, Lemma 2 − 5] studied **derivations** in ∗-**prime** **rings** and proved the following: Let R be a ∗-**prime** ring having nonzero ∗-ideal I then (i) If d is a nonzero derivation on R which commutes with ∗ and [x, R]Id(x) = {0} for all x ∈ I, then R is commutative. (ii) If d is a nonzero derivation on R which commutes with ∗ and [d(x), x] = 0 for all x ∈ I , then R is commutative. (iii) Let d be a derivation of R satisfying d∗ = ± ∗ d. If d 2 (I) = {0}, then d = 0. (iv) Let d

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Proof. Let d be the associated derivation of F. By Theorem 2.6, we conclude that either dI 0 or R is commutative. If R is not commutative, then dI 0. Since R is a **prime** ring, dI 0 implies dR 0 and hence Fxy Fxy for all x, y ∈ R. Set Gx Fx ∓ x for all x ∈ R. Then Gxy Gxy for all x ∈ R. Now, our assumption Fx ◦ y ±x ◦ y gives Fxy Fyx ±xy yx, that is, Gxy Gyx 0 for all x, y ∈ I. Thus using Gxy −Gyx, we have Gxyz −Gyxz Gxzy Gxzy, that is, Gxy, z 0 for all x, y, z ∈ I. Thus 0 GII, I GIRI, I GIRI, I. Since R is **prime**, this implies GI 0 or I is commutative. By Lemma 2.1, I commutative implies that R is commutative, a contradiction. Therefore, GI 0 and hence Gx Fx ∓ x 0 for all x ∈ I.

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