On derivation of semiprime rings (original) (raw)

A note on generalized derivations of prime and semiprime rings

2018

In this manuscript, we investigate the differential identities on (semi)-prime rings involving generalized derivations. Further, we obtain the structure of rings and information about the form of generalized derivations on prime rings in terms of the multiplication by the specific element from the extended centroid 1 Preliminaries and Motivation Throughout this paper, R is a (semi)-prime ring with the center Z(R), Q is the Martindale quotient ring of R and U is the Utumi quotient ring of R. The center of U denoted by C is called the extended centroid of R. For more details we refer to the reader [3]. For any x, y ∈ R, the symbol [x, y] and x◦y denote the commutator xy−yx and anti-commutator xy+yx respectively. Given x, y ∈ R we set x◦0y = x, x◦1y = x◦y = xy+yx, and inductively x◦my = (x◦m−1y)◦y for m > 1. Let us remind some basic notations and definitions for the sake of completeness. A ring R is said to be prime if xRy = (0) implies that x = 0 or y = 0 and R is semiprime ring if...

On Generalized Derivations of Semiprime Rings

International Journal of Pure and Apllied Mathematics, 2014

Let F be a commuting generalized derivation, with associated derivation d, on a semiprime ring R. We show that d(x)[y, z] = 0 for all x, y, z ∈ R and d is central. We define and characterize dependent elements of F and investigate a decomposition of R relative to F .

A note on a pair of derivations of semiprime rings

International Journal of Mathematics and Mathematical Sciences, 2004

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.

Remarks on Semiprime Rings with Generalized Derivations

2012

Let R be a ring with centre Z(R). An additive mapping F : R −→ R is said to be a generalized derivation if there exists a derivation d : R −→ R such that F(xy )= F(x)y+xd(y), for all x,y ∈ R (the map d is called the derivation associated with F). In the present note we prove that if a semiprime ring R admits a generalized derivation F, d is the nonzero associated derivation of F, satisfying certain polynomial constraints on a nonzero ideal I, then R contains a nonzero central ideal. Mathematics Subject Classification: Primary 16N60; Secondary 16W25

ON MULTIPLICATIVE (GENERALIZED)-DERIVATIONS IN SEMIPRIME RINGS

C o m m u n. Fa c. S c i. U n iv. A n k. S é r. A 1 M a th. S ta t. Vo lu m e 6 6 , N u m b e r 1 , P a g e s 1 5 3 –1 6 4 (2 0 1 7) D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 8 4 IS S N 1 3 0 3 –5 9 9 1 Abstract. In this paper, we study commutativity of a prime or semiprime ring using a map F : R ! R, multiplicative (generalized)-derivation and a map H : R ! R; multiplicative left centralizer, under the following conditions: For all x; y 2 R, i) F (xy) H(xy) = 0; ii) F (xy) H(yx) = 0; iii) F (x)F (y) H(xy) = 0; iv) F (xy) H(xy) 2 Z; v) F (xy) H(yx) 2 Z; vi) F (x)F (y) H(xy) 2 Z.

Orthogonal Generalized Symmetric Bi-Derivations of Semiprime Rings

Contemporary Mathematics and Statistics, 2017

The purpose of this paper is to study the notation of orthogonal symmetric generalized bi-derivations in semiprime rings and we proved orthogonality results. Let (∆ 1 , B 1) and (∆ 2 , B 2) be two generalized symmetric bi-derivations of R, then the following conditions are equivalent: (i) (∆ 1 , B 1) and (∆ 2 , B 2) are orthogonal. (ii) For all x, y, z ∈ R, the following relations hold (a) ∆ 1 (x, y)∆ 2 (y, z)+ ∆ 2 (x, y)∆ 1 (y, z) = 0. (b) B 1 (x, y)∆ 2 (y, z) + B 2 (x, y)∆ 1 (y, z) = 0. (iii) ∆ 1 (x, y)∆ 2 (y, z) = B 1 (x, y)∆ 2 (y, z) = 0 , for all x, y, z ∈ R. (iv) ∆ 1 (x, y)∆ 2 (y, z) = 0 , for all x, y, z ∈ R and B 1 ∆ 2 = B 1 B 2 = 0. (v) (∆ 1 ∆ 2 , B 1 B 2) is a generalized bi-derivation and ∆ 1 (x, y)∆ 2 (y, z) = 0, for all x, y, z ∈ R.

On a pair of (α, β)-derivations of semiprime rings

Aequationes mathematicae, 2005

A note on power values of derivation in prime and semiprime rings

Let R be a ring with derivation d, such that (d(xy))^n =(d(x))^n(d(y))^n for all x,y in R and n>1 is a fi?xed integer. In this paper, we show that if R is a prime, then d = 0 or R is a commutative. If R is a semiprime, then d maps R in to its center. Moreover, in semiprime case let A = O(R) be the orthogonal completion of R and B = B(C) be the Boolian ring of C, where C is the extended centroid of R, then there exists an idempotent e in B such that eA is commutative ring and d induce a zero derivation on (1-e)A.

On derivation of semiprime rings (original) (raw)

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