Murray and von Neumann 1 and von Neumann 2 introduced the notion of free action on abelian von Neumann algebras and used it for construction of certain factors. Kallman 3 generalized the notion of free action of automorphisms to von Neumann algebras, not necessarily abelian, by using implicitly the **dependent** **elements** of an automorphism. **Dependent** **elements** of automorphisms were later studied by Choda et al. 4 in the context of C ∗ -algebras. Several other authors have studied **dependent** **elements** of automorphisms in the context of operator algebras see 5, 6 and references therein. A brief account of **dependent** **elements** in W ∗ -algebras has also appeared in the book of Str˘atil˘a 7.

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The purpose of this paper is to investigate identities with **derivations** and automorphisms on **semiprime** **rings**. A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. Mayne proved that in case there exists a nontrivial centralizing automorphism on a prime ring, then the ring is commutative. In this paper, some results related to Posner’s theorem as well as to Mayne’s theorem are proved.

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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Abstract. Let R be a 2-torsion free ring and L a Lie ideal of R. An additive mapping F : R → R is called a generalized derivation on R if there exists a derivation d : R → R such that F(xy) = F (x)y+xd(y) holds for all x, y ∈ R. In the present paper we describe the action of generalized **derivations** satisfying several conditions on Lie ideals of **semiprime** **rings**.

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We study certain properties of **derivations** on **semiprime** **rings**. The main purpose is to prove the following result: let R be a **semiprime** ring with center Z(R), and let f , g be **derivations** of R such that f (x)x + xg(x) ∈ Z(R) for all x ∈ R, then f and g are central. As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear **derivations** satisfying the above central property. We also show that every skew-centralizing derivation f of a **semiprime** ring R is skew-commuting.

Abstract: Let n ≥ 2 be any fixed positive integer and ä denote the trace of symmetric skew reverse n-derivation Ä :R n → R, associated with an antiautomorphism á * .Let I be any Ideal of R.(1)If R is non commutative prime ring such that [ ä (x), á * (x)]=0, for all x I then Ä = 0 in R.(2)Let R be non commutative **semiprime** ring such that ä is commuting on I

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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Let R be a ring with center Z and I a nonzero ideal of R. An additive mapping F : R → R is called a generalized derivation of R if there exists a derivation d : R → R such that Fxy Fxy xdy for all x, y ∈ R. In the present paper, we prove that if Fx, y ±x, y for all x, y ∈ I or Fx ◦ y ±x ◦ y for all x, y ∈ I , then the **semiprime** ring R must contains a nonzero central ideal, provided d I / 0. In case R is prime ring, R must be commutative, provided d / 0. The cases i Fx, y ± x, y ∈ Z and ii Fx ◦ y ± x ◦ y ∈ Z for all x, y ∈ I are also studied.

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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A well-know result of Herstein [11] states that every Jordan isomorphism on a prime ring of characteristic di ﬀ erent from 2 is either an isomorphism or an anti-isomorphism. We remark that the situation where the **rings** are **semiprime** **rings** does not hold. In the same time, he showed that every Jordan derivation on a prime ring of characteristic dif- ferent from 2 is a derivation [12]. A brief proof of this result can be found in [4]. This result is extended by [3, 8] to the **semiprime** case.

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.

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].

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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Hvala 1 first introduced the generalized **derivations** in **rings** and obtained some remarkable results in classical **rings**. Daif and Tammam El-Sayiad 2 studied the generalized **derivations** in **semiprime** **rings**. The authors consider an additive mapping G : R → R of a ring R with the property Gx 2 GxxxDx for some derivation D of R. They prove that G is a Jordan

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for all x, y ∈ U . Let M (C) be the set of all maximal ideals of C and P ∈ M (C). Now by the standard theory of orthogonal completions for **semiprime** **rings** (see [24, p.31-32]), we have P U is a prime ideal of U invariant under all **derivations** of U . Moreover, T {P U | P ∈ M (C) } = 0. Set U = U/P U. Then derivation d canonically induces a derivation d on U defined by d(x) = d(x) for all x ∈ U . Therefore,

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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

Now, the following lemmas of orthogonal symmetric higher bi-**derivations** on Г- ring M. Lemma (2-3):(2) Let M be a 2-torsion free **semiprime** Г-ring an x,y be **elements** of M. If for all 𝛼, 𝛽𝜖Г, then the following conditions are equivalent:

1. W.E.Barnes, On the Γ -**rings** of Nobusawa, Pacific J. Math., 18 (1966), 411-422. 2. Y.Ceven, Jordan left **derivations** on completely prime gamma **rings**, C.U. Fen-Edebiyat Fakultesi, Fen Bilimleri Dergisi, (2002) Cilt 23 Sayi 2. 3. M.F.Hoque and A.C.Paul, On Centralizers of **Semiprime** Gamma **Rings**, International Mathematical Forum, 6(13) (2011) 627-638.