A NOTE ON Α-Derivations ON SEMIPRIME RINGS

Thaheem, A. B.; Samman, Mohammad

doi:10.1515/dema-2001-0406

Cited by 8 publications

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“…This research has been motivated by the work of Brešar [11], Lee [20] and Thaheem [29]. Throughout, R will represent an associative ring with center Z(R).…”

Section: Introductionmentioning

confidence: 99%

“…A mapping f : R → R is called skew-centralizing on R if f (x)x + xf (x) ∈ Z(R) holds for all x ∈ R; in particular, if f (x)x+xf (x) = 0 is fulfilled for all x ∈ R, then it is called skewcommuting on R. Brešar [7] has proved that if R is a 2-torsion free semiprime ring, and f : R → R is an additive skew-commuting mapping on R, then f = 0. Thaheem [29] has proved that in case D, G is a pair of derivations on a semiprime ring R satisfying the equation D(x)x + xG(x) = 0 for all x ∈ R, then D and G map R into Z(R) and G = −D. Let us point out that the equation of the type f (x)x + xg(x) = 0 for a pair of operators f and g on von Neumann algebras and C * −algebras appears in operator theory; in particular, in the study of elementary operators and other operator equations (see [30] and references therein for a detailed account of elementary operators and other operator equations).…”

Section: Introductionmentioning

confidence: 99%

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Identities with derivations on rings and Banach algebras

Vukman¹

2005

Glas. Mat. Ser. III

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Abstract. In this paper we prove the following result. Let m ≥ 1, n ≥ 1 be integers and let R be a 2mn(m + n − 1)!−torsion free semiprime ring. Suppose there exist derivations D, G : R → R such that D(x m )x n + x n G(x m ) = 0 holds for all x ∈ R. In this case both derivations D and G map R into its center and D = −G. We apply this purely algebraic result to obtain a range inclusion result of continuous derivations on Banach algebras.

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“…This research has been motivated by the work of Brešar [11], Lee [20] and Thaheem [29]. Throughout, R will represent an associative ring with center Z(R).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Identities with derivations on rings and Banach algebras

Vukman¹

2005

Glas. Mat. Ser. III

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“…(2) The corollary above was proved by Thaheem and Samman [23] for the case where m = n = 1 and by Vukman [24] with the additional assumption that R is 2mn(m + n − 1)!-torsion free.…”

Section: Proposition 34 a Derivation δ Of A Semiprime Ring R Is X-imentioning

confidence: 96%

An Identity With Generalized Derivations

Lee

Zhou

2009

J. Algebra Appl.

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Communicated by M. FerreroLet R be a prime ring that is not commutative and such that R ∼ = M 2 (GF(2)), let D, G be two generalized derivations of R, and let m, n be two fixed positive integers. Then D(x m )x n = x n G(x m ) for all x ∈ R iff the following two conditions hold: (1) There exists w ∈ Q, the symmetric Martindale quotient ring of R, such that D(x) = xw and G(x) = wx for all x ∈ R; (2) either w ∈ C, or x m and x n are C-dependent for all x ∈ R. We also consider the situation for the semiprime case. . Downloaded from www.worldscientific.com by PENNSYLVANIA STATE UNIVERSITY on 03/15/15. For personal use only.= bx − xb is a derivation of R, which is called the inner derivation induced by the element b. In [4] Brešar introduced the algebraic definition of generalized derivations. A map G : R → R is called a generalized derivation if there exists a derivation δ : R → R such thatfor all x, y ∈ R. The derivation δ is uniquely determined by the generalized derivation G. We call δ the derivation associated with G. For a, b ∈ R, the map x ∈ R → ax + xb defines a generalized derivation, which is called an inner generalized derivation. It is well-known that each derivation (resp. generalized derivation) of R can be uniquely extended to a derivation (resp. generalized derivation) of U . A derivation (resp. generalized derivation) of R is called X-inner if its extension to U is inner. Otherwise, it is called X-outer. Inner generalized derivations have been extensively studied in the theory of operator algebras (see [1, 4, 2, 22, etc.]). The study of generalized derivations from the viewpoint of pure algebra was given in [5, 13, 18, etc.]. In this paper we will prove the following Theorem 1.1. Let R be a prime ring that is not commutative and such that R ∼ = M 2 (GF(2)), let D, G be two generalized derivations of R, and let m, n be two fixed positive integers. Then D(x m )x n = x n G(x m ) for all x ∈ R iff the following two conditions hold: (1) There exists w ∈ Q such that D(x) = xw and G(x) = wx for all x ∈ R; (2) either w ∈ C, or x m and x n are C-dependent for all x ∈ R. Remarks.(1) We notice that the exceptional case in Theorem 1.1 indeed occurs. For instance, let w = 0 1 1 1 ∈ M 2 (GF(2)). Then x 3 wx = xwx 3 for all x ∈ M 2 (GF(2)), but w 3 and w are independent over GF(2).(2) We must characterize prime rings R satisfying the property (*): There exist distinct positive integers n, m such that x m and x n are C-dependent for all x ∈ R. A complete description is given as follows:Let R be a prime ring, not commutative, with extended centroid C. Then ( * ) holds iff C is a finite field and R ∼ = M s (C) for some integer s > 1.Proof. "⇒" Since (*) holds, x n yx m = x m yx n for all x, y ∈ R. Thus R is a prime PI-ring (as n = m). By Posner's Theorem [14, p. 57], Z(R), the center of R, is nonzero, and RC is a finite-dimensional central simple C-algebra, where C is equal to the quotient field of Z(R). Clearly, x m and x n are C-dependent for all x ∈ RC. The Wedderburn-Artin Theorem asserts that RC ∼ = M s (D), where...

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“…Inspired by the works of Brešar [5,6] and Thaheem [9] and the above remarks regarding generalized inner derivations, we consider a general situation regarding a pair of derivations of a semiprime ring and prove the following. Let f , g be a pair of derivations of a semiprime ring R satisfying f (x)x + xg(x) ∈ Z(R) for all x ∈ R, then f and g are central (Theorem 2.2).…”

Section: Introduction and Preliminaries Throughout R Denotes A Ringmentioning

confidence: 99%

A note on a pair of derivations of semiprime rings

Chaudhry

Thaheem

2004

International Journal of Mathematics and Mathematical Sciences

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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.2000 Mathematics Subject Classification: 47A50, 47B50. 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 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).We now recall some facts concerning semiprime rings and their extended centroids. For any semiprime ring R, one can construct the ring of quotients Q of R [1]. As R can be embedded isomorphically in Q, we consider R as a subring of Q. If the element q ∈ Q commutes with every element in R, then q belongs to C, the center of Q. C contains the centroid of R and is called the extended centroid of R. In general, C is a von Neumann regular ring, and it is a field if and only if R is a prime [1, Theorem 5]. For more information on extended centroid of R, we refer to [2].Brešar [6, Theorem 2] has proved that if R is a prime ring of characteristic not 2 and f : R → R is an additive skew-commuting mapping (i.e., f satisfies f (x)x + xf (x) = 0 for all x ∈ R), then f = 0.Moreover, Brešar [5, Theorem 4.1] has considered a pair of derivations on a prime ring and has proved the following. Let R be a prime ring and U a nonzero left ideal of R. Suppose that the derivations d and g of R are such that d(u)u − ug(u) ∈ Z(R) for all u ∈ U . If d ≠ 0, then R is commutative.

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A NOTE ON Α-Derivations ON SEMIPRIME RINGS

Cited by 8 publications

References 8 publications

Identities with derivations on rings and Banach algebras

Identities with derivations on rings and Banach algebras

An Identity With Generalized Derivations

A note on a pair of derivations of semiprime rings

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