Let R be an associative ring. The center of R is denoted by Z. For x, y ∈ R, the symbol x, y will denote the commutator xy − yx and the symbol x ◦ y will denote the anticommutator xy yx. We will make extensive use of basic commutator identities xy, z x, zy xy, z, x, yz x, yz yx, z. An additive mapping d from R to R is called a derivation of R if dxy dxy xdy holds for all x, y ∈ R. An additive mapping g from R to R is called a **generalized** derivation of R if there exists a derivation d from R to R such that gxy gxy xdy holds for all x, y ∈ R. Obviously, every derivation is a **generalized** derivation of R. Thus, **generalized** derivation covers both the concept of derivation and left multiplier mapping. A mapping F from R to R is called centralizing on S where S ⊆ R, if Fx, x ∈ Z for all x ∈ S.

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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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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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Lemma 1.2. Let R be a **prime** ring of characteristic different from 2 and L be a Lie ideal of R. Suppose R admits a nonzero **generalized** derivation (F, d) such that F(x)[x, y] = 0 (or [x, y]F (x) = 0) for all x, y ∈ L, then L ⊆ Z(R). Proof. Suppose by contradiction that L is not central in R. By [11] (pages 4-5) there exists a non-central ideal I of R such that 0 6= [I, R] ⊆ L. By our

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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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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 3.2. Let be a **prime** -ring with 2-torsion free and a nonzero ideal of . Let : → be a **generalized** derivation and left **generalized** derivation of , with associated derivation d on . If ∈ and [(), ] & = 0, for all ∈ , ∈ Γ, then either ∈ () or '() = 0.

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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A mapping h : R → R defined by h(x) = ax + xb (x ∈ R) for some a, b ∈ R is called a **generalized** inner derivation [8]. **Generalized** inner **derivations** are called elementary operators and have been extensively studied in operator algebras. We note that the condition that h is centralizing on R can be written in the form [a, x]x + x[b, x] ∈ Z(R) for all x ∈ R. Thus, introducing inner **derivations** f and g by f (x) = [a, x] and g(x) = [b, x], we obtain the condition as in [5, Theorem 4.1], that is, f (x)x + xg(x) ∈ Z(R) for all x ∈ R.

We investigate identities with **derivations** and automorphisms on **semiprime** **rings**. We prove, for example, that in case there exist a derivation D : R → R and an automorphism α : R → R, where R is a 2-torsion-free **semiprime** ring, such that [D(x)x + xα(x),x] = 0 holds for all x ∈ R, then D and α − I , where I denotes the identity mapping, map R into its center. Throughout this paper, R will represent an associative ring with center Z(R). A ring R is 2-torsion-free in case 2x = 0 implies that x = 0 for any x ∈ R. As usual, we write [x, y] for xy − yx and make use of the commutator identities [xy, z] = [x,z]y + x[y,z], [x, yz] = [x, y]z + y[x,z], x, y,z ∈ R. We denote by I the identity mapping of a ring R. An additive mapping D : R → R is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R. Let α be an automorphism of a ring R. An additive map- ping D : R → R is called an α-derivation if D(xy ) = D(x)α(y) + xD(y) holds for all pairs x, y ∈ R. Note that the mapping D = α − I is an α-derivation. Of course, the concept of α-derivation generalizes the concept of derivation, since I -derivation is a derivation. We denote by C the extended centroid of a **semiprime** ring R and by Q Martindale ring of quotients. For the explanation of the extended centroid of a **semiprime** ring R and the Martindale ring of quotients, we refer the reader to [1]. A mapping f of R into itself is called centralizing on R if [ f (x), x] ∈ Z(R) holds for all x ∈ R; in the special case when [ f (x),x] = 0 holds for all x ∈ R, the mapping f is said to be commuting on R. The history of commuting and centralizing mappings goes back to 1955 when Divinsky [5] proved that a simple Artinian ring is commutative if it has a commuting nontrivial automor- phism. Two years later, Posner [8] has proved that the existence of a nonzero centralizing derivation on a **prime** ring forces the ring to be commutative (Posner’s second theorem). Luh [6] **generalized** the Divinsky result, we have just mentioned above, to arbitrary **prime**

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Lemma 3.10. Let N be a **prime** Γ-near-ring, f a nonzero **generalized** derivation of N associated with the nonzero derivation d and a ∈ N. i If aΓfN 0, then a 0. ii If fNΓa 0, then a 0. Proof. i For any x, y ∈ N, α, β ∈ Γ, we get 0 aβf xαy aβf xαyaβxαdy aβxαdy. Hence aΓNΓdN 0. Since N is a **prime** Γ-near-ring and d / 0, we obtain a 0.

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Theorem 4: Let be a **prime** ring and be a nonzero ideal of . If admits a left **generalized** derivation associated with a nonzero derivation such that ( ) + = 0holds for all , ∈ , then is commutative. Proof: For any , ∈ , we have ( ) + = 0. If = 0, then = 0, for all , ∈ . Replacing by and using the fact that = − , we find that [ , ] = 0 for all , , ∈ and hence [ , ] = (0) for all , ∈ . Since

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.

It was shown in [6] that symmetric biderivations are related to general solution of some functional equations. The notion of additive commuting mappings is closely connected with the notion of biderivations. Every commuting additive mapping f : R −→ R gives rise to a bideriva- tion on R. Namely linearizing [x, f (x)] = 0 for all x, y ∈ R, (x, y) 7→ [ f (x), y] is a biderivation. Typical examples are mapping of the form (x, y) 7−→ λ [x, y] for all x, y ∈ R, where λ ∈ C, the extended centroid of R. We shall call such maps inner biderivations. There has been ongo- ing interest concerning the relationship between the commutativity of a ring and the existence of certain specific types of **derivations**. Some results on symmetric biderivation in **prime** and **semiprime** **rings** can be found in [1, 5, 8].

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Lemma 2.8. Let N be 3-**prime** near-ring. If N admits a left **generalized** multiplicative derivation f with associated multiplicative derivation d such that f (u)v = uf (v) for all u, v ∈ N , then d = 0. Proof. We are given that f (u)v = uf (v) for all u, v ∈ N . Now replacing v by vw, where w ∈ N , in the previous relation, we obtain that f (u)vw = uf (vw) i.e.; f (u)vw = u(vf (w) + d(v)w). By using hypothesis we arrive at ud(v)w = 0 i.e.; uN d(v)w = {0}. Now using the facts that N ƒ= {0} and N is a 3-**prime** near-ring, we obtain that d(v)w = 0, for all v, w ∈ N . This shows that d(v)w = 0 i.e.; d(N )N w = {0}. Again 3-primeness of N and N ƒ= {0} force us to conclude that d(N ) = {0}. We get d = 0.

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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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Theorem 3.7 Let ring R be **prime** with𝑐ℎ𝑎𝑟 𝑅 ≠ 2, (0) ≠ 𝐿 be a square closed Lie ideal of 𝑅 and 0 ≠ 𝐹, 𝐺: 𝑅 → 𝑅 are **generalized** 𝛼 − **derivations** associated with 𝛼 − **derivations** 𝑑 and 𝑔 respectively such that 𝑠 ∈ 𝑍(𝐿)|0 ≠ 𝑑 𝑠 , 0 ≠ 𝑔 𝑠 ∈ 𝑍 𝐿 , 𝑑 𝑠 ≠ ∓𝑔 𝑠 ≠ ∅ and 𝛼 𝐿 ⊂ 𝐿. If one of the following conditions is satisfy, then 𝐿 ⊆ 𝑍 𝑅 .

derivation ݀ on a **prime** Γ -ring ܯ for which ܯ is commutative with the conditions ݀(ܷ) ⊆ ܷ and ݀ ଶ (ܷ) ⊆ ܼ , where ܷ is an ideal of ܯ and ܼ is the centre of ܯ . Sapanci and Nakajima [10] defined a derivation and a Jordan derivation on Γ -**rings** and investigated a Jordan derivationon a certain type of completely **prime** Γ -ring which is a derivation. They proved that every Jordan derivation on a 2 -torsion free completely **prime** Γ -**rings** is a derivation. They also gave examples of a derivation and a Jordan derivation of Γ -**rings**. Rahman and Paul [3,4] worked on Jordan **derivations** and Jordan left **derivations** of 2 and 3-torsion free **semiprime** Γ -**rings**. Rahman and Paul [5,6] have also worked **derivations** and **generalized** **derivations** on Lie ideals of completely **semiprime** Γ -**rings**. Reddy, Nagesh and Kumar [7] and Reddy, Kumar and Reddy [8] worked on left **generalized** **derivations** on **prime** γ-**rings** and results of symmetric reverse bi-**derivations** on **prime** **rings** respectively Kadhim et al. [11] studied on higher characterization on Γ -ring called Gamma * derivation pair and jordan gamma*− derivation pair on gamma-ring M with involution.

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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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R is commutative or char(R) ≠ 2 and R satisﬁes the standard identity of degree 4. Several authors have studied what happens if the Engel condition is satisﬁed by the elements of a nonzero one-sided ideal of R. To be more speciﬁc, in [2] Bell and Martin- dale proved that if R is **semiprime** and [[d(x 1 ), x 1 ], x 2 ] = 0, for all x 1 in a nonzero left