# Top PDF Remarks on Generalized Derivations in Prime and Semiprime Rings ### Remarks on Generalized Derivations in Prime and Semiprime Rings

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. ### On Prime and Semiprime Rings with Additive Mappings and Derivations

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 . 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 (, , , , where further references can be found). In , the description of all centralizing additive maps of a prime ring R of characteristic not 2 was given and subsequently in  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  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., , , , , , , , , , , , , , , , , ,  and ). The main objective of the present paper is to investigate commutativity of prime and semiprime rings satisfying certain identities involving additive mappings and 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. ### Lie Ideals and Generalized Derivations in Semiprime Rings

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  (pages 4-5) there exists a non-central ideal I of R such that 0 6= [I, R] ⊆ L. By our ### 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 ### Annihilator condition of a pair of derivations in prime and semiprime rings

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 ### Left Generalized Derivations on Prime Γ-Rings

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 dependent elements in rings

on F . It is easy to see that in semiprime rings there are no nonzero nilpotent dependent elements (see ). This fact will be used throughout the paper without speciﬁc refer- ences. Dependent elements were implicitly used by Kallman  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. . 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 . The purpose of this paper is to investigate dependent elements of some mappings related to derivations and automorphisms on prime and semiprime rings. ### A note on a pair of derivations of semiprime rings

A mapping h : R → R defined by h(x) = ax + xb (x ∈ R) for some a, b ∈ R is called a generalized inner derivation . 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. ### Identities with derivations and automorphisms on semiprime rings

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 . 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  proved that a simple Artinian ring is commutative if it has a commuting nontrivial automor- phism. Two years later, Posner  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  generalized the Divinsky result, we have just mentioned above, to arbitrary prime ### On Prime Gamma Near Rings with Generalized Derivations

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. ### Left Generalized Derivations and Commutativity of Prime Rings

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 ### Dependent Elements of Derivations on Semiprime Rings

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. ### Strong commutativity preserving biderivations on prime rings

It was shown in  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]. ### Optimized Multiplicative Derivations in Close to Rings

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. ### On Commutativity of *-Prime Near-Rings with Derivations

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 , 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 ### ClosedLieIdeals of Prime Rings with Generalizedα−derivations

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 . ### Lie Ideals on Prime Γ-Rings with Jordan Right Derivations

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  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  and Reddy, Kumar and Reddy  worked on left generalized derivations on prime γ-rings and results of symmetric reverse bi-derivations on prime rings respectively Kadhim et al.  studied on higher characterization on Γ -ring called Gamma * derivation pair and jordan gamma*− derivation pair on gamma-ring M with involution.  