# Derivations and generalized derivations

## Top PDF Derivations and generalized derivations: ### Lie Ideals and Generalized Derivations in Semiprime Rings

A well known result of Posner  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. ### Generalized Derivations in Semiprime Gamma Rings

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 ### Generalized Derivations on Prime Near Rings

Throughout the paper, 𝑁 denotes a zero-symmetric left near ring with multiplicative center 𝑍, and for any pair of elements 𝑥, 𝑦 ∈ 𝑁, [𝑥, 𝑦] denotes the commutator 𝑥𝑦 − 𝑦𝑥, while the symbol (𝑥, 𝑦) denotes the additive commutator 𝑥+𝑦−𝑥−𝑦. A near ring 𝑁 is called zero-symmetric if 0𝑥 = 0, for all 𝑥 ∈ 𝑁 (recall that left distributivity yields that 𝑥0 = 0). The near ring 𝑁 is said to be 3-prime if 𝑥𝑁𝑦 = {0} for 𝑥, 𝑦 ∈ 𝑁 implies that 𝑥 = 0 or 𝑦 = 0. A near ring 𝑁 is called 2-torsion-free if (𝑁, +) has no element of order 2. A nonempty subset 𝐴 of 𝑁 is called a semigroup right (resp., semigroup left) ideal if 𝐴𝑁 ⊆ 𝐴 (resp., 𝑁𝐴 ⊆ 𝐴), and if 𝐴 is both a semigroup right ideal and a semigroup left ideal, it is called a semigroup ideal. An additive mapping 𝑑 : 𝑁 → 𝑁 is a derivation on 𝑁 if 𝑑(𝑥𝑦) = 𝑑(𝑥)𝑦+ 𝑥𝑑(𝑦) , for all 𝑥, 𝑦 ∈ 𝑁 . An additive mapping 𝑓 : 𝑁 → 𝑁 is said to be a right (resp., left) generalized derivation with associated derivation 𝑑 if 𝑓(𝑥𝑦) = 𝑓(𝑥)𝑦 + 𝑥𝑑(𝑦) (resp., 𝑓(𝑥𝑦) = 𝑑(𝑥)𝑦 + 𝑥𝑓(𝑦)), for all 𝑥, 𝑦 ∈ 𝑁, and 𝑓 is said to be a generalized derivation with associated derivation 𝑑 on 𝑁 if it is both a right generalized derivation and a left generalized derivation on 𝑁 with associated derivation 𝑑. (Note that this definition differs from the one given by Hvala in ; his generalized derivations are our right generalized derivations.) Every derivation on 𝑁 is a generalized derivation. ### Generalized derivations in prime rings

Theorem 2.1. Let R be a prime ring and I be a nonzero ideal of R. Suppose that F, G, H : R → R are generalized derivations, associated with derivations d, g, h : R → R respectively and α : R → R is any map such that F (xy) + G(x)H(y) + [α(x), y] = 0, for all x, y ∈ I. If g, h are non zero derivation then R is commutative. ### Left Generalized Derivations and Commutativity of Prime Rings

The study of the commutativity of prime rings with derivations was initiated by E.C.Posner. Hvala  studied generalized derivations in prime rings. Daif and Bell  established that if in a semiprime ring there exists a nonzero ideal of and a derivation such that ([ , ]) = [ , ] for all , ∈ , then ⊆ ( ). Jaya Subba Reddy et.al studied centralizing and commuting left generalized derivation on prime ring is commutative. J.H. Mayne  studied on centralizing mappings of prime rings. M.A.Quadriet.al 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. ### Hyers Ulam Rassias stability of generalized derivations

Let Ꮽ be an algebra and let ᐄ be an Ꮽ-bimodule. A linear mapping μ : Ꮽ → ᐄ is called a generalized derivation if there exists a derivation (in the usual sense) δ : Ꮽ → ᐄ such that μ(ab) = aμ(b) + δ(a)b for all a,b ∈ Ꮽ. Familiar examples are the derivations from Ꮽ to ᐄ and all so-called inner generalized derivations; those are defined by μ x,y (a) = xa − ay for ### On commutativity of near rings with generalized derivations

The study of commutativity of 3-prime near-rings by using derivations was initiated by Bell and Mason in 1987 in . After that, Beidar, Fong and Wang generalize some results from rings to near-rings with derivations in . Again, Bell generalizes several results of  by using one (two) sided semigroup ideal of the near-ring in his work in . In 2006, Golbasi used in  and  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  possible analogues of three results of  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  and  and prove some theorems of commutativity for 3-prime near-rings. ### Stability of Homomorphisms and Generalized Derivations on Banach Algebras

where 0 < p < 1 is a fixed parameter and f : I → R is unknown, I is a nonvoid open interval and 2.1 holds for all x, y ∈ I. They characterized the equivalence of 2.1 and Jensen’s functional equation in terms of the algebraic properties of the parameter p. For p 1/2 in 2.1, we get the Jensen’s 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 → Y, ### On Prime Gamma Near Rings with Generalized Derivations

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. ### Left Generalized Derivations on Prime Γ-Rings

The notion of -ring was first introduction by Nobusawa  and also shown that -ring, more general than rings. Barnes  slightly weakened the conditions in the definitions of a -rings in the sense of Nobusawa. After the study of -rings by Nobusawa  and Barnes , many researchers have a done lot of work and have obtained some generalizations of the corresponding results in ring theory . Barnes  and kyuno  studied the structure of -ring and obtained various generalizations of the corresponding results of ring theory. Hvala  introduced the concept of Generalized derivations in rings. Dey, Paul and Rakhimov  discussed some properties of Generalized derivations in semiprime gamma rings Bresar  studied on the distance of the composition of two derivations to the generalized derivations. Jaya Subba Reddy. et al.  studied centralizing and commutating left generalized derivation on prime ring is commutative. Jaya Subba Reddy et al.  studied some results of symmetric reverse bi-derivations on prime rings, Ozturk et al.  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. ### Generalized Derivations and Bilocal Jordan Derivations of Nest Algebras

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 12–15. In fact, under appropriate conditions, local derivations are derivations. ### Generalized derivations with power values in rings and Banach algebras

In , 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 ∈ U and a derivation d on U . ### Left Annihilator of Identities Involving Generalized Derivations in Prime Rings

Remark 1.6. In , Lee extended the definition of generalized derivation as follows: by a generalized derivation we mean an additive mapping F : I → U such that F(xy) = F (x)y + xd(y) holds for all x, y ∈ 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 ∈ U and a derivation d on U. ### Remarks on Generalized Derivations in Prime and Semiprime Rings

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. ### Identities in \$3\$-prime near-rings with left multipliers

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 , , , , , , , where further references can be found. In , 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  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  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 , , , , where more references can be found). Recently the second author together with Ali  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 ### Superstability of Generalized Derivations

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

additive function F : R → R is called a generalized derivation if there exists a derivation d : R → R such that F(xy) = F(x)y + xd(y) holds for all x, y ∈ R. In this paper, we prove that d = 0 or U ⊆ Z(R) if any one of the following conditions holds: (1) d(x) ◦ F(y) = 0, (2) [d(x), F(y) = 0], (3) either d(x) ◦ F(y) = x ◦ y or d(x) ◦ F(y) + x ◦ y = 0, (4) either d(x) ◦ F(y) = [x, y] or d(x) ◦ F (y) + [x, y] = 0, (5) either d(x) ◦ F(y) − xy ∈ Z(R) or d(x) ◦ F (y) + xy ∈ 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 ◦ y or [d(x), F(y)] + x ◦ y = 0 for all x, y ∈ U. ### A Note on Generalized Skew Derivations on Rings

Throughout the paper R will represent an associative ring with center Z(R). For any x,yϵR the symbol [x,y] will denote the commutator xy-yx; while the symbol xᵒy will stand for anti-commutator xy+yx. R is prime if aRb=0 implies a=0 or b=0. An additive map *:R→R is called an involution if * is an anti-automorphism of order 2; that is (x*)*=x for all xϵR. 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)⊆H(R), otherwise it is said to be of second kind. In the later case Z(R)⋂S(R)⧧(0). An additive mapping d:R→R is said to be a derivation if d(xy)=d(x)y+xd(y) for all x,y ϵR. An additive map F:R→R is a generalized derivation if their exists a derivation d such that F(xy)=F(x)y+xd(y) for all x,yϵR. All derivations are generalized derivations. ### (1,α)- Derivations in Prime г - near Rings

The notion of a ring a concept more general than a ring was definedby Nobusawa . As a generalization of near rings, rings were introduced by satyanarayana. The derivation of a near ring has been introduced by Bell and Mason. 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. 