An Identity With Generalized Derivations

Abstract

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...

Strong commutativity preserving generalized derivations on Lie ideals

“…maps of type x°ax þ xb for some a, b 2 R). The notion of generalized derivations was introduced by Bresˇar [7] and these maps had been extensively studied in various directions (see, for instance, [6,15,16,20,21,24,26,28,[30][31][32][33]36,37]). An additive subgroup L of R is said to be a Lie…”

Identities with Generalized Derivations on Prime Rings and Banach Algebras

“…If in the previous theorem, we consider the whole ring R as a Lie ideal, we get the main result of [18] in the case m = n and precisely we get: Theorem 2. Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C = Z(U ) the extended centroid of R, H and G nonzero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that H(x n )x n + x n G(x n ) = 0 for all x ∈ R, then either R is commutative or there exists a ∈ U such that H(x) = xa and G(x) = −ax.…”

“…More recently, in [18] Lee and Zhou considered a similar situation where the derivations are replaced by generalized derivations. More specifically, an additive map G : R → R is said to be a generalized derivation if there is a derivation d of R such that for all x, y ∈ R, G(xy) = G(x)y + xd(y).…”

“…[18] 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 if and only if the following two conditions hold: (i) There exists w ∈ Q, the symmetric Martindale quotient ring of R, such that D(x) = xw and G(x) = wx for all x ∈ R; (ii) either w ∈ C, or x m and x n are C-dependent for all x ∈ R.…”

Strong commutativity preserving generalized derivations on right ideals

“…A map g : R → R is called a generalized derivation of R if there exists a derivation d of R such that g(x + y) = g(x) + g(y) and g(x y) = g(x)y + xd (y) for all x, y ∈ R. Basic examples are derivations and generalized inner derivations (i.e., maps of type x → ax + xb for some a, b ∈ R). The notion of generalized derivations was introduced by Brešar [8] and these maps had been extensively studied in ring theory and operator algebras (see for instance [1,7,14,17,18,21,22,24,26,30,31,33,[36][37][38] (1) R is commutative;…”