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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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 β 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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Throughout the paper, π denotes a zero-symmetric left near ring with multiplicative center π, and for any pair of elements π₯, π¦ β π, [π₯, π¦] denotes the commutator π₯π¦ β π¦π₯, while the symbol (π₯, π¦) denotes the additive commutator π₯+π¦βπ₯βπ¦. A near ring π is called zero-symmetric if 0π₯ = 0, for all π₯ β π (recall that left distributivity yields that π₯0 = 0). The near ring π is said to be 3-prime if π₯ππ¦ = {0} for π₯, π¦ β π implies that π₯ = 0 or π¦ = 0. A near ring π is called 2-torsion-free if (π, +) has no element of order 2. A nonempty subset π΄ of π is called a semigroup right (resp., semigroup left) ideal if π΄π β π΄ (resp., ππ΄ β π΄), and if π΄ is both a semigroup right ideal and a semigroup left ideal, it is called a semigroup ideal. An additive mapping π : π β π is a derivation on π if π(π₯π¦) = π(π₯)π¦+ π₯π(π¦) , for all π₯, π¦ β π . An additive mapping π : π β π is said to be a right (resp., left) **generalized** derivation with associated derivation π if π(π₯π¦) = π(π₯)π¦ + π₯π(π¦) (resp., π(π₯π¦) = π(π₯)π¦ + π₯π(π¦)), for all π₯, π¦ β π, and π is said to be a **generalized** derivation with associated derivation π on π if it is both a right **generalized** derivation and a left **generalized** derivation on π with associated derivation π. (Note that this definition differs from the one given by Hvala in [1]; his **generalized** **derivations** are our right **generalized** **derivations**.) Every derivation on π is a **generalized** derivation.

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Theorem 2.1. Let R be a prime ring and I be a nonzero ideal of R. Suppose that F, G, H : R β R are **generalized** **derivations**, associated with **derivations** d, g, h : R β R respectively and Ξ± : R β R is any map such that F (xy) + G(x)H(y) + [Ξ±(x), y] = 0, for all x, y β I. If g, h are non zero derivation then R is commutative.

The study of the commutativity of prime rings with **derivations** was initiated by E.C.Posner[12]. Hvala [8] studied **generalized** **derivations** in prime rings. Daif and Bell [6] established that if in a semiprime ring there exists a nonzero ideal of and a derivation such that ([ , ]) = [ , ] for all , β , then β ( ). Jaya Subba Reddy et.al[14] studied centralizing and commuting left **generalized** derivation on prime ring is commutative. J.H. Mayne [11] studied on centralizing mappings of prime rings. M.A.Quadriet.al[13] studied **generalized** **derivations** and commutativity of prime ring is a commutative. In this paper we extended some results on left **generalized** **derivations** and commutativity of prime ring is a commutative.The proofs of the following results can be seen in [1, Lemma 3] and [11, Lemma 3] respectively.

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Let α be an algebra and let α be an α-bimodule. A linear mapping ΞΌ : α β α is called a **generalized** derivation if there exists a derivation (in the usual sense) Ξ΄ : α β α such that ΞΌ(ab) = aΞΌ(b) + Ξ΄(a)b for all a,b β α. Familiar examples are the **derivations** from α to α and all so-called inner **generalized** **derivations**; those are defined by ΞΌ x,y (a) = xa β ay for

The study of commutativity of 3-prime near-rings by using **derivations** was initiated by Bell and Mason in 1987 in [4]. After that, Beidar, Fong and Wang generalize some results from rings to near-rings with **derivations** in [1]. Again, Bell generalizes several results of [4] by using one (two) sided semigroup ideal of the near-ring in his work in [2]. In 2006, Golbasi used in [8] and [9] the definition that a map on a near-ring is called a **generalized** derivation if it is both a left and a right **generalized** derivation on the near-ring. Golbasi used that definition to deduce some results on the class of 3-prime near-rings. In 2008, Bell investigated in [3] possible analogues of three results of [4] for **generalized** **derivations** on near-rings and he got only one extension. Recently, many papers studied the commutativity on near-rings by using **derivations** and **generalized** **derivations**, such as [6, 7, 10, 11]. In this paper, we generalize some results of [3] and [11] and prove some theorems of commutativity for 3-prime near-rings.

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where 0 < p < 1 is a fixed parameter and f : I β R is unknown, I is a nonvoid open interval and 2.1 holds for all x, y β I. They characterized the equivalence of 2.1 and Jensenβs functional equation in terms of the algebraic properties of the parameter p. For p 1/2 in 2.1, we get the Jensenβs functional equation. In the present paper, we establish the general solution and some stability results concerning the functional equation 2.1 in normed spaces for p 1/3. This applied to investigate and prove the **generalized** Hyers-Ulam stability of homomorphisms and **generalized** **derivations** in real Banach algebras. In this section, we assume that X is a normed algebra and Y is a Banach algebra. For convenience, we use the following abbreviation for a given mapping f : X β Y,

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

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The notion of -ring was first introduction by Nobusawa [9] and also shown that -ring, more general than rings. Barnes [1] slightly weakened the conditions in the definitions of a -rings in the sense of Nobusawa. After the study of -rings by Nobusawa [9] and Barnes [1], many researchers have a done lot of work and have obtained some generalizations of the corresponding results in ring theory [6][8]. Barnes [1] and kyuno [8] studied the structure of -ring and obtained various generalizations of the corresponding results of ring theory. Hvala [4] introduced the concept of **Generalized** **derivations** in rings. Dey, Paul and Rakhimov [3] discussed some properties of **Generalized** **derivations** in semiprime gamma rings Bresar [2] studied on the distance of the composition of two **derivations** to the **generalized** **derivations**. Jaya Subba Reddy. et al. [5] studied centralizing and commutating left **generalized** derivation on prime ring is commutative. Jaya Subba Reddy et al. [12] studied some results of symmetric reverse bi-**derivations** on prime rings, Ozturk et al. [10] studied on **derivations** of prime gamma rings. Khan et al. [6,7] studied on **derivations** and **generalized** **derivations** on prime - rings is a commutative. In this paper we extended some results on left **generalized** **derivations** on prime -ring is a commutative.

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Based on a great deal of research works of many mathematicians, some scholars paid more interests in similar kind of problems under more **generalized** conditions, such as considering local **derivations** on nest algebras and **generalized** derivation. Zhu and Xiong 6, 7 proved that local **derivations** of nest algebra and standard operator algebra are **derivations**, Zhang 8 considered the Jordan **derivations** of nest algebras, Lee 9 discussed **generalized** **derivations** of left faithful rings. Recently, some scholars discussed some new types of **derivations**, as Li and Zhou 10 and Majieed and Zhou 11 investigated some new types of **generalized** **derivations** associated with Hochschild 2 cocycles, other examples are in 12β15. In fact, under appropriate conditions, local **derivations** are **derivations**.

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In [23], Lee proved that every **generalized** derivation can be uniquely extended to a general- ized derivation of U and thus all **generalized** **derivations** of R implicitly assumed to be defined on the whole of U. In particular, Lee proved the follwing: Let R be a semiprime ring. Then every **generalized** derivation F on a dense right ideal of R can be uniquely extended to U and assumes the form F(x) = ax + d(x) for some a β U and a derivation d on U .

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Remark 1.6. In [23], Lee extended the definition of **generalized** derivation as follows: by a **generalized** derivation we mean an additive mapping F : I β U such that F(xy) = F (x)y + xd(y) holds for all x, y β I, where I is a dense left ideal of R and d is a derivation from I into U . Moreover, Lee also proved that every **generalized** derivation can be uniquely extended to a **generalized** derivation of U , and thus all **generalized** **derivations** of R will be implicitly assumed to be defined on the whole of U. Lee obtained the following: every **generalized** derivation F on a dense left ideal of R can be uniquely extended to U and assumes the form F (x) = ax + d(x) for some a β U and a derivation d on U.

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Let R be a ring with center Z and I a nonzero ideal of R. An additive mapping F : R β R is called a **generalized** derivation of R if there exists a derivation d : R β R such that Fxy Fxy xdy for all x, y β R. In the present paper, we prove that if Fx, y Β±x, y for all x, y β I or Fx β¦ y Β±x β¦ y for all x, y β 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 Β± x, y β Z and ii Fx β¦ y Β± x β¦ y β Z for all x, y β I are also studied.

with associated derivation d if F (xy) = F (x)y + xd(y), for all x, y β 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 β 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 β 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 β 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]) β [x, y] = 0 for all x, y β I or d([x, y]) + [x, y] = 0 for all x, y β 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]) Β± [x, y] β Z(R) for all x, y β 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]) β [x, y] β Z(R) for all x, y β I, or (ii)F (x β¦ y) + x β¦ y β Z(R) for all x, y β 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 β R such that F ([x, y]) = [x, y] with F 6= Id R

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The well-known problem of stability of functional equations started with a question of Ulam 1 in 1940. In 1941, Ulamβs problem was solved by Hyers 2 for Banach spaces. Aoki 3 provided a generalization of Hyersβ theorem for approximately additive mappings. In 1978, Rassias 4 **generalized** Hyersβ theorem by obtaining a unique linear mapping near an approximate additive mapping.

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additive function F : R β R is called a **generalized** derivation if there exists a derivation d : R β R such that F(xy) = F(x)y + xd(y) holds for all x, y β R. In this paper, we prove that d = 0 or U β Z(R) if any one of the following conditions holds: (1) d(x) β¦ F(y) = 0, (2) [d(x), F(y) = 0], (3) either d(x) β¦ F(y) = x β¦ y or d(x) β¦ F(y) + x β¦ y = 0, (4) either d(x) β¦ F(y) = [x, y] or d(x) β¦ F (y) + [x, y] = 0, (5) either d(x) β¦ F(y) β xy β Z(R) or d(x) β¦ F (y) + xy β Z(R), (6) either [d(x), F(y)] = [x, y] or [d(x), F(y)] + [x, y] = 0, (7) either [d(x), F(y)] = x β¦ y or [d(x), F(y)] + x β¦ y = 0 for all x, y β U.

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Throughout the paper R will represent an associative ring with center Z(R). For any x,yΟ΅R the symbol [x,y] will denote the commutator xy-yx; while the symbol xα΅y will stand for anti-commutator xy+yx. R is prime if aRb=0 implies a=0 or b=0. An additive map *:RβR is called an involution if * is an anti-automorphism of order 2; that is (x*)*=x for all xΟ΅R. R is *-prime if aRb*=0 implies a=0 or b=0. An element x in a ring R with involution * is said to be hermitian and skew-hermitian elements of R will be denoted by H(R) and S(R), respectively. The involution is said to be of the firest kind if Z(R)βH(R), otherwise it is said to be of second kind. In the later case Z(R)βS(R)β§§(0). An additive mapping d:RβR is said to be a derivation if d(xy)=d(x)y+xd(y) for all x,y Ο΅R. An additive map F:RβR is a **generalized** derivation if their exists a derivation d such that F(xy)=F(x)y+xd(y) for all x,yΟ΅R. All **derivations** are **generalized** **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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The study of commutativity of 3-prime near-rings was initiated by using **derivations** by H.E. Bell and G. Mason [6] in 1987. Subsequently a number of authors have investigated the commutativity of 3-prime near-rings admitting different types of **derivations**, **generalized** **derivations**, **generalized** multiplicative **derivations**( for reference see [1, 3, 4, 5, 6, 7, 8, 9], where further references can be found). In the present paper, we have obtained the commutativity of 3-prime near-rings, equipped with left **generalized** multiplicative **derivations** and satisfying some differential identities or conditions.

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