# Top PDF On Prime and Semiprime Rings with Additive Mappings and Derivations

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

for all x, y ∈ I. Right multiplying (11) by y and then subtracting from (12), we have F(x)[F(y), y] = 0 for all x, y ∈ I. Putting xz for x in the last relation, we obtain F (x)z[F (y), y] = 0 for all x, y, z ∈ I. This implies [F (x), x]z[F (x), x] = 0 for all x, z ∈ I, that is, [F (x), x]I[F(x), x] = (0) for all x ∈ I. Since R is prime, it follows that [F(x), x] = 0 for all x ∈ I. Then by Lemma 2.5, F (x) = λx + ζ(x) for all x ∈ I, where λ ∈ C and ζ : I → C. Since F is additive, ζ is also additive map. Moreover, since F is left multiplier, so the map ζ(x) = F(x) − λx is also a left multiplier on I. This implies that F(x) = λx + ζ(x) for all x ∈ I. Hence, we obtain our conclusion (1). Theorem 3.4. Let R be a semiprime ring, I a nonzero ideal of R, and let F, d : R → R be two additive mappings such that F(xy) = F (x)y + xd(y) for all x, y ∈ R. If F (x)F (y) ± yx ∈ Z (R) for all x, y ∈ I, then [d(x), x] = 0 for all x ∈ I. Moreover, if d is a derivation such that d(I) ̸ = (0), then R contains a nonzero central ideal.

### 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

### Symmetric Skew Reverse n-Derivations on Prime Rings and Semiprime rings

Theorem.3.1. Let n ≥2 be any fixed positive integer. Let R be a non commutative n! – torsion free prime ring and I be any non – zero two sided ideal of R .Suppose that there exist a symmetric skew reverse n-derivation Ä:R n →R associated with an antiautomorphism á *. Let ä denote the trace of Ä such that [ä(x), á * (x)] = 0, for all x  I then

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

Proof. Let d be the associated derivation of F. By Theorem 2.6, we conclude that either dI 0 or R is commutative. If R is not commutative, then dI 0. Since R is a prime ring, dI 0 implies dR 0 and hence Fxy Fxy for all x, y ∈ R. Set Gx Fx ∓ x for all x ∈ R. Then Gxy Gxy for all x ∈ R. Now, our assumption Fx ◦ y ±x ◦ y gives Fxy Fyx ±xy yx, that is, Gxy Gyx 0 for all x, y ∈ I. Thus using Gxy −Gyx, we have Gxyz −Gyxz Gxzy Gxzy, that is, Gxy, z 0 for all x, y, z ∈ I. Thus 0 GII, I GIRI, I GIRI, I. Since R is prime, this implies GI 0 or I is commutative. By Lemma 2.1, I commutative implies that R is commutative, a contradiction. Therefore, GI 0 and hence Gx Fx ∓ x 0 for all x ∈ I.

### A note on a pair of derivations of semiprime rings

1. Introduction and preliminaries. Throughout, R denotes a ring with center Z(R). We write [x, y] for xy − yx. We will frequently use the identities [xy, z] = x[y, z] + [x, z]y and [x, yz] = y[x, z] + [x, y]z for all x, y, z ∈ R. We recall that R is semiprime if aRa = (0) implies a = 0 and it is prime if aRb = (0) implies a = 0 or b = 0. A prime ring is semiprime but the converse is not true in general. An additive mapping d : R → R is called a derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R. A mapping f : R → R is called centralizing if [f (x), x] ∈ Z(R) for all x ∈ R; in particular, if [f (x), x] = 0 for all x ∈ R, then it is called commuting. A mapping f : R → R is called central if f (x) ∈ Z(R) for all x ∈ R. Every central mapping is obviously commuting but not conversely, in general. A lot of work has been done on centralizing mappings (see, e.g., [3, 4, 5] and the references therein). A mapping f : R → R is called skew-centralizing if f (x)x + xf (x) ∈ Z(R) for all x ∈ R; in particular, if f (x)x + xf (x) = 0 for all x ∈ R, then it is called skew-commuting. We denote the radical of a Banach algebra A by rad(A).

### 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.

### Lie Ideals and Generalized Derivations in Semiprime Rings

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.

### Left Annihilator of Identities Involving Generalized Derivations in Prime Rings

Many results in literature indicate that global structure of a prime ring R is often tightly connected to the behavior of additive mappings defined on R. Asma, Rehman, Shakir in [1] proved that if d is a derivation of a 2-torsion free prime ring R which acts as a homomorphism or anti-homomorphism on a non-central Lie ideal of R such that u 2 ∈ L, for all u ∈ L, then d = 0. At this

### Identities with derivations and automorphisms on semiprime rings

Proof. The proof is an immediate consequence of Theorems 6 and 4. Corollary 8. Let R be a noncommutative prime ring of characteristic di ﬀ erent from two. Suppose that there exist a derivation D : R → R and an automorphism α : R → R, such that the mapping x → D(x) + α(x) is centralizing on R. In this case, D = 0 and α = I.

### On Skew Left n-Derivations with Lie Ideal Structure

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.

### (1,α)- Derivations in Prime г - near Rings

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.

### \$\Gamma\$*-Derivation Pair and Jordan \$\Gamma\$*-Derivation Pair on \$\Gamma\$-ring M with Involution

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].

### 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 [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

### On dependent elements in rings

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.

### *-Left Derivations on Prime Left Nearrings

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.

### 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.

### 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 [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

### On (?,?)-Derivations and Commutativity of Prime and Semi prime ?-rings Afrah Mohammad Ibraheem

Let R be a Г-ring, and σ, τ be two automorphisms of R. An additive mapping d from a Γ-ring R into itself is called a (σ,τ)-derivation on R if d(aαb) = d(a)α σ(b) + τ(a)αd(b), holds for all a,b ∈R and α∈Γ. d is called strong commutativity preserving (SCP) on R if [d(a), d(b)] α = [ ] ( ) holds for all a,b∈R and α∈Γ. In this paper, we

### ClosedLieIdeals of Prime Rings with Generalizedα−derivations

From [5], an additivemapping𝑑 from 𝑅 into 𝑅 is called derivation if 𝑑 𝑟𝑝 = 𝑑 𝑟 𝑝 + 𝑟𝑑 𝑝 for all 𝑟, 𝑝 ∈ 𝑅. In [3], Bresar introduced definition of generalized derivation. An additive mapping 𝐹 from 𝑅 into 𝑅 is called generalized derivation associated with derivation 𝑑 if 𝐹 𝑟𝑝 = 𝐹 𝑟 𝑝 + 𝑟𝑑 𝑝 for all 𝑟, 𝑝 ∈ 𝑅. From [1] and [4], definition of 𝛼 −derivation and generalized 𝛼 −derivation is given as follows: Let 𝑑 from 𝑅 into 𝑅 be an additive mapping and 𝛼 be an automorphism of 𝑅. If 𝑑 𝑟𝑝 = 𝑑 𝑟 𝑝 + 𝛼 𝑟 𝑑 𝑝 holds for all 𝑟, 𝑝 ∈ 𝑅, then 𝑑 is called 𝛼 −derivation. Let 𝐹 from 𝑅 into 𝑅 be an additive mapping. If 𝐹 𝑟𝑝 = 𝐹 𝑟 𝑝 + 𝛼 𝑟 𝑑 𝑝 holds for all 𝑟, 𝑝 ∈ 𝑅, then 𝐹 is called generalized 𝛼 −derivation associated with 𝛼 −derivation 𝑑. In many years, authors have proved commutative theorems for prime rings with 𝛼 − derivation and generalized 𝛼 − derivation. Also many researchershavegeneralizedresultstoidealsandLieideals of ring.