# Top PDF Dependent Elements of Derivations on Semiprime Rings

### Dependent Elements of Derivations on Semiprime Rings

Murray and von Neumann 1 and von Neumann 2 introduced the notion of free action on abelian von Neumann algebras and used it for construction of certain factors. Kallman 3 generalized the notion of free action of automorphisms to von Neumann algebras, not necessarily abelian, by using implicitly the dependent elements of an automorphism. Dependent elements of automorphisms were later studied by Choda et al. 4 in the context of C ∗ -algebras. Several other authors have studied dependent elements of automorphisms in the context of operator algebras see 5, 6 and references therein. A brief account of dependent elements in W ∗ -algebras has also appeared in the book of Str˘atil˘a 7.

### Identities with derivations and automorphisms on semiprime rings

The purpose of this paper is to investigate identities with derivations and automorphisms on semiprime rings. A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. Mayne proved that in case there exists a nontrivial centralizing automorphism on a prime ring, then the ring is commutative. In this paper, some results related to Posner’s theorem as well as to Mayne’s theorem are proved.

### 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 [28]. 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 ([10], [12], [15], [25], where further references can be found). In [14], the description of all centralizing additive maps of a prime ring R of characteristic not 2 was given and subsequently in [4] 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 [12] 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., [1], [2], [3], [5], [6], [7], [8], [9], [16], [17], [18], [19], [20], [24], [25], [27], [29], [31] and [32]). The main objective of the present paper is to investigate commutativity of prime and semiprime rings satisfying certain identities involving additive mappings and derivations.

### Lie Ideals and Generalized Derivations in Semiprime Rings

Abstract. Let R be a 2-torsion free ring and L a Lie ideal of R. An additive mapping F : R → R is called a generalized derivation on R if there exists a derivation d : R → R such that F(xy) = F (x)y+xd(y) holds for all x, y ∈ R. In the present paper we describe the action of generalized derivations satisfying several conditions on Lie ideals of semiprime rings.

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

We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: let R be a semiprime ring with center Z(R), and let f , g be derivations of R such that f (x)x + xg(x) ∈ Z(R) for all x ∈ R, then f and g are central. As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivation f of a semiprime ring R is skew-commuting.

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

Abstract: Let n ≥ 2 be any fixed positive integer and ä denote the trace of symmetric skew reverse n-derivation Ä :R n → R, associated with an antiautomorphism á * .Let I be any Ideal of R.(1)If R is non commutative prime ring such that [ ä (x), á * (x)]=0, for all x  I then Ä = 0 in R.(2)Let R be non commutative semiprime ring such that ä is commuting on I

### 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

### 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.

### 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.

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

A well-know result of Herstein [11] states that every Jordan isomorphism on a prime ring of characteristic di ﬀ erent from 2 is either an isomorphism or an anti-isomorphism. We remark that the situation where the rings are semiprime rings does not hold. In the same time, he showed that every Jordan derivation on a prime ring of characteristic dif- ferent from 2 is a derivation [12]. A brief proof of this result can be found in [4]. This result is extended by [3, 8] to the semiprime case.

### 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.

### \$\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].

### (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.

### 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

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

for all x, y ∈ U . Let M (C) be the set of all maximal ideals of C and P ∈ M (C). Now by the standard theory of orthogonal completions for semiprime rings (see [24, p.31-32]), we have P U is a prime ideal of U invariant under all derivations of U . Moreover, T {P U | P ∈ M (C) } = 0. Set U = U/P U. Then derivation d canonically induces a derivation d on U defined by d(x) = d(x) for all x ∈ U . Therefore,

### 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

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

Now, the following lemmas of orthogonal symmetric higher bi-derivations on Г- ring M. Lemma (2-3):(2) Let M be a 2-torsion free semiprime Г-ring an x,y be elements of M. If for all 𝛼, 𝛽𝜖Г, then the following conditions are equivalent:

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

1. W.E.Barnes, On the Γ -rings of Nobusawa, Pacific J. Math., 18 (1966), 411-422. 2. Y.Ceven, Jordan left derivations on completely prime gamma rings, C.U. Fen-Edebiyat Fakultesi, Fen Bilimleri Dergisi, (2002) Cilt 23 Sayi 2. 3. M.F.Hoque and A.C.Paul, On Centralizers of Semiprime Gamma Rings, International Mathematical Forum, 6(13) (2011) 627-638.