for all x, y ∈ I. Right multiplying (11) by y and then subtracting from (12), we have F(x)[F(y), y] = 0 for all x, y ∈ I. Putting xz for x in the last relation, we obtain F (x)z[F (y), y] = 0 for all x, y, z ∈ I. This implies [F (x), x]z[F (x), x] = 0 for all x, z ∈ I, that is, [F (x), x]I[F(x), x] = (0) for all x ∈ I. Since R is **prime**, it follows that [F(x), x] = 0 for all x ∈ I. Then by Lemma 2.5, F (x) = λx + ζ(x) for all x ∈ I, where λ ∈ C and ζ : I → C. Since F is **additive**, ζ is also **additive** map. Moreover, since F is left multiplier, so the map ζ(x) = F(x) − λx is also a left multiplier on I. This implies that F(x) = λx + ζ(x) for all x ∈ I. Hence, we obtain our conclusion (1). Theorem 3.4. Let R be a **semiprime** ring, I a nonzero ideal of R, and let F, d : R → R be two **additive** **mappings** such that F(xy) = F (x)y + xd(y) for all x, y ∈ R. If F (x)F (y) ± yx ∈ Z (R) for all x, y ∈ I, then [d(x), x] = 0 for all x ∈ I. Moreover, if d is a derivation such that d(I) ̸ = (0), then R contains a nonzero central ideal.

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Let R be a **prime** ring, U the Utumi quotient ring of R, C = Z(U ) the extended centroid of R, L a non-central Lie ideal of R, H and G non-zero generalized **derivations** of R. Suppose that there exists an integer n ≥ 1 such that H(u n )u n + u n G(u n ) ∈ C, for all u ∈ L, then either there exists a ∈ U such that H(x) = xa, G(x) = −ax, or R satisfies the standard identity s 4 . Moreover, in the

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Theorem.3.1. Let n ≥2 be any fixed positive integer. Let R be a non commutative n! – torsion free **prime** ring and I be any non – zero two sided ideal of R .Suppose that there exist a symmetric skew reverse n-derivation Ä:R n →R associated with an antiautomorphism á *. Let ä denote the trace of Ä such that [ä(x), á * (x)] = 0, for all x I then

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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1. Introduction and preliminaries. Throughout, R denotes a ring with center Z(R). We write [x, y] for xy − yx. We will frequently use the identities [xy, z] = x[y, z] + [x, z]y and [x, yz] = y[x, z] + [x, y]z for all x, y, z ∈ R. We recall that R is **semiprime** if aRa = (0) implies a = 0 and it is **prime** if aRb = (0) implies a = 0 or b = 0. A **prime** ring is **semiprime** but the converse is not true in general. An **additive** mapping d : R → R is called a derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R. A mapping f : R → R is called centralizing if [f (x), x] ∈ Z(R) for all x ∈ R; in particular, if [f (x), x] = 0 for all x ∈ R, then it is called commuting. A mapping f : R → R is called central if f (x) ∈ Z(R) for all x ∈ R. Every central mapping is obviously commuting but not conversely, in general. A lot of work has been done on centralizing **mappings** (see, e.g., [3, 4, 5] and the references therein). A mapping f : R → R is called skew-centralizing if f (x)x + xf (x) ∈ Z(R) for all x ∈ R; in particular, if f (x)x + xf (x) = 0 for all x ∈ R, then it is called skew-commuting. We denote the radical of a Banach algebra A by rad(A).

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Throughout, R will represent an associative ring with centre ZR. The commutator xy − yx will be denoted by x, y. We will use the basic commutator identities xy, z x, zy xy, z and x, yz yx, z x, yz. Recall that a ring R is **semiprime** if aRa 0 implies a 0 and is **prime** if aRb 0 implies a 0 or b 0. An **additive** mapping d : R → R is called a derivation on R if dxy dxy xdy for all x, y ∈ R. It is called commuting if dx, x 0 for all x ∈ R. Let a ∈ R, then the mapping d : R → R given by dx a, x is a derivation on R. It is called inner derivation on R.

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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Many results in literature indicate that global structure of a **prime** ring R is often tightly connected to the behavior of **additive** **mappings** defined on R. Asma, Rehman, Shakir in [1] proved that if d is a derivation of a 2-torsion free **prime** ring R which acts as a homomorphism or anti-homomorphism on a non-central Lie ideal of R such that u 2 ∈ L, for all u ∈ L, then d = 0. At this

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Proof. The proof is an immediate consequence of Theorems 6 and 4. Corollary 8. Let R be a noncommutative **prime** ring of characteristic di ﬀ erent from two. Suppose that there exist a derivation D : R → R and an automorphism α : R → R, such that the mapping x → D(x) + α(x) is centralizing on R. In this case, D = 0 and α = I.

Several authors have studied the concept as commuting and centralizing **derivations** like J. Vukman who investigated symmetric bi-**derivations** on **prime** and **semiprime** **rings** (9). We obtain the similar results of Jung and Park ones for permuting 3-**derivations** on **prime** and **semiprime** **rings** (10) and more results in (11, 12, 13, 14, 15). In the present paper, we have introduced the notion of skew left 𝓃-derivation and skew left 𝓃-derivationn associated with the antiautomorphism 𝛼 ∗ and studied the commuting and centralizing of this notion and commutativity of Lie ideal under certain conditions.

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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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Barnes [3], Luh [13], Kyuno [11], Hoque and Paul [8] as well as Uddin and Islam [16,17] studied the structure of Г-**rings** and obtained various generalizations of corresponding parts in ring theory. Note that during the last few decades, many authors have studied **derivations** in the context of **prime** and **semiprime** **rings** and Γ-**rings** with involution [1,2,4,10,18].The notion of derivation pair and Jordan derivation pair on a *- ring R were defined by [12, 14, 19,20].

with associated derivation d if F (xy) = F (x)y + xd(y), for all x, y ∈ N and F is called a left generalized derivation with associated derivation d if F(xy) = d(x)y + xF (y), for all x, y ∈ N . F is called a generalized derivation with associated derivation d if it is both a left as well as a right generalized derivation with associated derivation d. An **additive** mapping F : N → N is said to be a left (resp. right) multiplier (or centralizer) if F(xy) = F(x)y (resp. F(xy) = xF(y)) holds for all x, y ∈ N . F is said to be a multiplier if it is both left as well as right multiplier. Notice that a right (resp. left) generalized derivation with associated derivation d = 0 is a left (resp. right) multiplier. Several authors investigated the commutativity in **prime** and **semiprime** **rings** admitting **derivations** and generalized **derivations** which satisfy appropriate algebraic conditions on suitable subset of the **rings**. For example, we refer the reader to [1], [3], [11], [12], [15], [18], [19], where further references can be found. In [11], Daif and Bell proved that if R is a **prime** ring admitting a derivation d and I a nonzero ideal of R such that d([x, y]) − [x, y] = 0 for all x, y ∈ I or d([x, y]) + [x, y] = 0 for all x, y ∈ I, then R is commutative. Further, Hongan [15] generalized the above result and proved that if R is a **semiprime** ring with a nonzero ideal I and d is a derivation of R such that d([x, y]) ± [x, y] ∈ Z(R) for all x, y ∈ I , then I is a central ideal. In particular, if I = R, then R is commutative. Recently, Dhara [12] generalized this result by replacing derivation d with a generalized derivation F in a **prime** ring R. More precisely, he proved that if R is a **prime** ring and I a nonzero ideal of R which admits a generalized derivation F associated with a nonzero derivation d such that either (i)F ([x, y]) − [x, y] ∈ Z(R) for all x, y ∈ I, or (ii)F (x ◦ y) + x ◦ y ∈ Z(R) for all x, y ∈ I, then R is commutative. There has been a great deal of work concerning left (or right) multiplier in **prime** or **semiprime** **rings** (see for reference [4], [16], [17], [21], where more references can be found). Recently the second author together with Ali [4] proved that if a **prime** ring R admits a left multiplier F : R → R such that F ([x, y]) = [x, y] with F 6= Id R

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on F . It is easy to see that in **semiprime** **rings** there are no nonzero nilpotent dependent elements (see [11]). This fact will be used throughout the paper without speciﬁc refer- ences. Dependent elements were implicitly used by Kallman [10] to extend the notion of free action of automorphisms of abelian von Neumann algebras of Murray and von Neumann [14, 17]. They were later on introduced by Choda et al. [8]. Several other au- thors have studied dependent elements in operator algebras (see [6, 7]). A brief account of dependent elements in W ∗ -algebras has been also appeared in the book of Str˘ atil˘ a [16]. The purpose of this paper is to investigate dependent elements of some **mappings** related to **derivations** and automorphisms on **prime** and **semiprime** **rings**.

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The recent results on **rings** deal with commutativity of **prime** and **semiprime** **rings** admitting suitably constrained **derivations**. It is natural to look for comparable results on nearrings and this has been done in [3],[4],[5],[6],[2], and [1].It is our purpose to extend some of these results on **prime** nearrings admitting suitably constrained ( , )*- derivation.

The **derivations** in Γ-near-**rings** have been introduced by Bell and Mason 1. They studied basic properties of **derivations** in Γ-near-**rings**. Then As¸ci 2 obtained commutativity conditions for a Γ-near-ring with **derivations**. Some characterizations of Γ-near-**rings** and regularity conditions were obtained by Cho 3. Kazaz and Alkan 4 introduced the notion of two-sided Γ-α-derivation of a Γ-near-ring and investigated the commutativity of a **prime** and **semiprime** Γ-near-**rings**. Uc¸kun et al. 5 worked on **prime** Γ-near-**rings** with **derivations** and they found conditions for a Γ-near-ring to be commutative. In 6 Dey et al. studied commutativity of **prime** Γ-near-ring with generalized **derivations**.

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Throughout the present paper, unless otherwise mentioned, N will denote a left near- ring. N is called a **prime** near-ring if xN y = {0} implies x = 0 or y = 0. It is called **semiprime** if xN x = {0} implies x = 0. Given an integer n > 1, near-ring N is said to be n-torsion free, if for x ∈ N , nx = 0 implies x = 0. If K is a nonempty subset of N, then a normal subgroup (K, +) of (N, +) is called a right ideal (resp. a left ideal) of N if (x + k)y − xy ∈ K (resp. xk ∈ K) holds for all x, y ∈ N and for all k ∈ K. K is called an ideal of N if it is both a left ideal as well as a right ideal of N . The symbol Z will denote the multiplicative center of N , that is, Z = {x ∈ N | xy = yx for all y ∈ N}. For any x, y ∈ N the symbol [x, y] = xy − yx stands for multiplicative commutator of x and y, while the symbol xoy will represent xy + yx. For terminologies concerning near- **rings**, we refer to G.Pilz [1, 2]. Following [3], an **additive** mapping d : N −→ N satisfying d(xy) = xd(y) + d(x)y for all x, y ∈ N is called a derivation on N . A ∗-near ring N is called ∗-**prime** near-ring if xN y = xN y ∗ = {0} implies that either x = 0 or y = 0. Let N be a ∗-near-ring. An ideal I of N is called ∗-ideal if I ∗ = I. An element x ∈ N is called a symmetric element if x ∗ = x and an element x ∈ N is called a skewsymmetric element if x ∗ = −x. We denote the collection of all symmetric and skewsymmetric elements

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Let R be a Г-ring, and σ, τ be two automorphisms of R. An **additive** mapping d from a Γ-ring R into itself is called a (σ,τ)-derivation on R if d(aαb) = d(a)α σ(b) + τ(a)αd(b), holds for all a,b ∈R and α∈Γ. d is called strong commutativity preserving (SCP) on R if [d(a), d(b)] α = [ ] ( ) holds for all a,b∈R and α∈Γ. In this paper, we

From [5], an additivemapping𝑑 from 𝑅 into 𝑅 is called derivation if 𝑑 𝑟𝑝 = 𝑑 𝑟 𝑝 + 𝑟𝑑 𝑝 for all 𝑟, 𝑝 ∈ 𝑅. In [3], Bresar introduced definition of generalized derivation. An **additive** mapping 𝐹 from 𝑅 into 𝑅 is called generalized derivation associated with derivation 𝑑 if 𝐹 𝑟𝑝 = 𝐹 𝑟 𝑝 + 𝑟𝑑 𝑝 for all 𝑟, 𝑝 ∈ 𝑅. From [1] and [4], definition of 𝛼 −derivation and generalized 𝛼 −derivation is given as follows: Let 𝑑 from 𝑅 into 𝑅 be an **additive** mapping and 𝛼 be an automorphism of 𝑅. If 𝑑 𝑟𝑝 = 𝑑 𝑟 𝑝 + 𝛼 𝑟 𝑑 𝑝 holds for all 𝑟, 𝑝 ∈ 𝑅, then 𝑑 is called 𝛼 −derivation. Let 𝐹 from 𝑅 into 𝑅 be an **additive** mapping. If 𝐹 𝑟𝑝 = 𝐹 𝑟 𝑝 + 𝛼 𝑟 𝑑 𝑝 holds for all 𝑟, 𝑝 ∈ 𝑅, then 𝐹 is called generalized 𝛼 −derivation associated with 𝛼 −derivation 𝑑. In many years, authors have proved commutative theorems for **prime** **rings** with 𝛼 − derivation and generalized 𝛼 − derivation. Also many researchershavegeneralizedresultstoidealsandLieideals of ring.

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Let 𝑁 be a near ring. An **additive** mapping 𝑓 : 𝑁 → 𝑁 is said to be a right generalized (resp., left generalized) derivation with associated derivation 𝑑 on 𝑁 if 𝑓(𝑥𝑦) = 𝑓(𝑥)𝑦 + 𝑥𝑑(𝑦) (resp., 𝑓(𝑥𝑦) = 𝑑(𝑥)𝑦 + 𝑥𝑓(𝑦)) for all 𝑥, 𝑦 ∈ 𝑁. A mapping 𝑓 : 𝑁 → 𝑁 is said to be a generalized derivation with associated derivation 𝑑 on 𝑁 if 𝑓 is both a right generalized and a left generalized derivation with associated derivation 𝑑 on 𝑁. The purpose of the present paper is to prove some theorems in the setting of a semigroup ideal of a 3-**prime** near ring admitting a generalized derivation, thereby extending some known results on **derivations**.

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