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

### 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).

### 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

### Lie Ideals and Generalized Derivations in Semiprime Rings

Assume now that d(P ) * P , then d(P) 6= 0 and d(P )R 6= 0. Moreover note that, for any p ∈ P and r, s ∈ R, d(pr)s = d(p)rs + pd(r)s implies that d(P)R ⊆ d(P R)R + P, in particular d(P)R is a non-zero right ideal of R. Starting from our main assumption and linearizing we have that F(x)[z, y] + F (z)[x, y] = 0, for all x, y, z ∈ L. For any p ∈ P, r ∈ R, u ∈ L, replace x by [pr, u]. By computation it follows [v, u][z, y] = 0, for all v ∈ d(P)R and u, z, y ∈ L. By using the same argument of Lemma 1.2, since L is not central in R, one has that d(P )R is a central right ideal of R, which implies that R is commutative, a contradiction.

### 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 [1]. 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 [5] proved that a simple Artinian ring is commutative if it has a commuting nontrivial automor- phism. Two years later, Posner [8] 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 [6] generalized the Divinsky result, we have just mentioned above, to arbitrary prime

### A note on (α,β)*-i-n-Derivations in Rings with Involution

Let R be a ∗-ring. An additive mapping d : R −→ R is said to be a ∗-derivation on R if d(xy) = d(x)y ∗ + xd(y) holds for all x, y ∈ R. If R is a commutative ∗-ring, then d : R −→ R defined by d(x) = a(x− x ∗ ), where a ∈ R, is a ∗-derivation on R ( for reference see [10]). An additive map T : R −→ R is called a left (resp. right) ∗-multiplier if T (xy) = T (x)y ∗ (resp. T (xy) = x ∗ T (y)) holds for all x, y ∈ R. An additive mapping d : R −→ R is said to be a (α, β)-∗-derivation on R if d(xy) = d(x)α(y ∗ ) + β(x)d(y) holds for all x, y ∈ R. There has been a great deal of work concerning commutativity of prime and semiprime rings admitting different types of derivations ( for reference see [3 − 6],[10],[14] etc., where further references can be found). Very recently Ali and Khan [2] defined symmetric ∗- biderivation, symmetric left (resp. right) ∗-bimultiplier and studied some properties of prime ∗-rings and semiprime ∗-rings, admitting symmetric ∗-biderivation and symmetric left (resp. right) ∗-bimultiplier. Motivated by these concepts the authors [6] introduced the notion of ∗-n-derivations and ∗-n-multipliers in ∗-rings and studied these notions in the setting of prime ∗-rings and semiprime ∗-rings.

### A Note on Generalized Skew Derivations on Rings

Let R be an associative ring and α be an automorphism of R. An additive mapping D:R→R is called a skew- derivation of R if D(xy)=D(x)y+α(x)D(y) for all x,yϵR and α is called an associated automorphism of D. An additive mapping G:R→R is said to be a generalized skew-derivation of R if there exists a skew-derivation D of R with associated automorphism α such that G(xy)=G(x)y+α(x)D(y) for all x,yϵR. The definition of generalized skew-derivation is a unified notion of skew-derivation and generalized derivation, which are considered as classical additive mappings of non-associative algebras. The behaviour of these has been investigated by many researchers from various views, see [1-5]. In [6, Theorem 2], Daif and Bell proved that if R is a semiprime witha nonzero ideal I and d is a derivation of R such that d([x,y]=[x,y] for all x,yϵI, then I⊆Z(R). In particular if R is a prime ring, then R must be commutative. Recently in [7] Filippis and Huang studied the situation (F([x,y])) n =[x,y] for all x,yϵI, where I is a nonzero ideal in a prime ring R, F is a generalized derivation of R and n1, a fixed integer. In this case they conclude that either R is commutative or n=1: d=0 and F(x)=x for all xϵR.

### Higher derivations on rings and modules

of τ-torsion submodules of M. A torsion theory will be called hereditary if T is closed under submodules and it will be called cohereditary if F is closed under factor modules. Standard results and terminology on torsion theory can be found in [4, 10] while general information on rings and modules can be found in [2]. Finally, if N is a submodule of an R-module M, then for any x ∈ M, (N : x) will denote the right ideal of R given by { a ∈ R | xa ∈ N } and (0 : x) is the right ideal { a ∈ R | xa = 0 } .

### Generalized derivations in prime rings

In 2015, S.K Tiwari et al. [19] considered the following situations: (i) G(xy) ± F(x)F (y) ± xy ∈ Z (R); (ii) G(xy) ± F(x)F(y) ± yx ∈ Z(R); (iii) G(xy) ± F (y)F (x) ± xy ∈ Z(R);(iv) G(xy) ± F (y)F (x) ± yx ∈ Z(R); (v) G(xy) ± F (y)F (x) ± [x, y] ∈ Z(R); (vi) G(xy) ± F(x)F (y) ± [α(x), y] ∈ Z(R), for all x, y ∈ I, where I is a non-zero ideal in prime ring R, α : R → R is any mapping and F, G are two generalized derivations associated with derivations d, g respectively.

### Jordan Γ* Derivation on Semiprime Γ Ring M with Involution

The notion of Γ-ring was introduced as a generalized extension of the concept on classical ring. From its first appearance, the extensions and the generalizations of various important results in the theory of classical rings to the theory of Γ-rings have attracted a wider attention as an emerging field of research to enrich the world of al- gebra. A good number of prominent mathematicians have worked on this interesting area of research to develop many basic characterizations of Γ-rings. Nobusawa [1] first introduced the notion of a Γ-ring and showed that Γ-rings are more general than rings. Barnes [2] slightly weakened the conditions in the definition of Γ-ring in the sense of Nobusawa. Barnes [2], Luh [3], Kyuno [4], Hoque and Pual [5]-[7], Ceven [8], Dey et al. [9] [10], Vukman [11] and others obtained a large number of important basic properties of Γ-rings in various ways and developed more remarkable results of Γ-rings. We start with the following necessary introductory definitions.

### 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.

### 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.

### Jordan automorphisms, Jordan derivations of generalized triangular matrix algebra

We investigate Jordan automorphisms and Jordan derivations of a class of algebras called generalized triangular matrix algebras. We prove that any Jordan automorphism on such an algebra is either an automorphism or an antiautomorphism and any Jordan derivation on such an algebra is a derivation.

### Left Generalized Derivations on Prime Γ-Rings

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.

### Generalized Derivations of Prime Rings

In [3], Bergen et al. investigate the relationship between the derivations and Lie ideals of a prime ring, and obtain some useful results. In [4], P. H. Lee and T. K. Lee get six suﬃcient conditions of central Lie ideal, which extend some results of commutativity on a prime ring. Motivated by the above, in this paper, we extend M. Ashraf ’s results to a Lie ideal of a prime ring. Throughout this paper, R will be a prime ring and U will always denote a Lie ideal of R.

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

antiautomorphism of ℛ. A ring ℛ is said to be 𝓃- torsion free if 𝓃𝒶=0 with 𝒶∈ℛ then 𝒶=0, where 𝓃 is nonzero integer (1). For any 𝜐,𝛾∈ℛ, the commutator 𝜐𝛾-𝛾𝜐 is denoted by [𝜐, 𝛾] (2). Recall that a ring ℛ is said to be prime if 𝒶ℛ𝑏=0 implies that either 𝒶=0 or 𝑏=0 for all 𝒶, 𝑏∈ℛ (3) and it is semiprime if 𝒶ℛ𝒶=0 implies that 𝒶=0 for all 𝒶 ∈ ℛ (1). An additive mapping 𝜉:ℛ → ℛ is called a derivation if 𝜉(υγ)= 𝜉(𝜐)γ + 𝜐 ξ(γ) for all 𝜐, 𝛾∈ℛ (4), and it is called a skew derivation (𝛼 ∗ -

### An Equation Related to Centralizers in Semiprime Gamma Rings

Let M be a Γ -ring. Then an additive subgroup U of M is called a left (right) ideal of M if M Γ U ⊂ U ( U Γ M ⊂ U ). If U is both a left and a right ideal, then we say U is an ideal of M . Suppose again that M is a Γ -ring. Then M is said to be a 2-torsion free if 2 x = 0 implies x = 0 for all x ∈ M . An ideal P 1 of a Γ -ring M is said to be prime if for any ideals A and B of M , A Γ B ⊆ P 1 implies A ⊆ P 1 or B ⊆ P 1 . An ideal P 2 of a Γ -ring M is said to be semiprime if for any ideal U of M , U Γ U ⊆ P 2 implies U ⊆ P 2 . A Γ -ring M is said to be prime if a Γ M Γ b = (0) with a , b ∈ M , implies a = 0 or b = 0 and semiprime if

### On Symmetric Reverse Ɵ*-Bicentralizer in Semiprime *- Rings

Theorem 3.8 : Let R be a 2-torsion free semiprime *- ring , then every symmetric right Jordan * - Bicentralizer of R is a symmetric right reverse * - Bicentralizer . Where is an automorphism on R Lemma 3.9 : Let R be a *- ring with identity . Then a symmetric biadditive mapping T : R x R R is a symmetric left ( right ) reverse * - Bicentralizer if and only if T is of the form T(x , y) = a y * ) x * ) (T(x , y) = x * ) y * ) a ) for some fixed element a . Where is an automorphism on R .

### On derivations and commutativity in prime rings

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

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

Semiprime ring with Symmetric skew 3-reverse derivations”, Trans. on Math., 1 (2015), 11-15. [11] C. Jayasubba Reddy, V.Vijaya Kumar and S.Mallikarjuna Rao, “Prime ring with symmetric skew 4-reverse derivation”, Int. J. of Pure Algebra, 6 (2016), 251-254.

### Orthogonal Symmetric Higher bi-Derivations on Semiprime Г-Rings

Let M be a 2-torsion free semiprime Г- ring such that 𝑥𝛼𝑦𝛽𝑧 = 𝑥𝛽𝑦𝛼𝑧 for all α, βϵГ , x, y, zϵM and two symmetric higher bi-derivations 𝑑 𝑛 and 𝑔 𝑛 for all n𝜖N are orthogonal if and only if 𝑑 𝑛 (𝑥, 𝑦)𝛼𝑔 𝑛 (𝑦, 𝑧) + 𝑔 𝑛 (𝑥, 𝑦)𝛼𝑑 𝑛 (𝑦, 𝑧) = 0 for all