Derivations of rings and Banach algebras (original) (raw)

A note on derivations in rings and Banach algebras

Algebraic structures and their applications

Let R be a prime ring with U the Utumi quotient ring and Q be the Martindale quotient ring of R, respectively. Let d be a derivation of R and m, n be fixed positive integers. In this paper, we study the case when one of the following holds: (i) d(x) •n d(y)=x •m y (ii) d(x) •m d(y)=(d(x • y)) n for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on Banach algebras. 1. Introduction and preliminaries In all that follows, unless specifically stated otherwise, R will be an associative ring, Z(R) the center of R, Q its Martindale quotient ring and U its Utumi quotient ring. The center of U , denoted by C, is called the extended centroid of R (we refer the reader to [3, Chapter 2], for the definitions and related properties of these objects). For any x, y ∈ R, the symbol [x, y] and x • y stands for the commutator xy − yx and anti-commutator xy + yx, respectively. Given

On derivations in prime and semiprime rings with Banach algebras

International Journal of Algebra

In this article we study certain differential identities in prime and semiprime rings. In particular, we prove if a prime rings satisfies certain differential identity on a nonzero ideal of a ring R, then R is commutative. Finally, as an application we obtain some range inclusion results of continuous generalized derivations on non-commutative Banach algebras.

Generalized derivations in prime rings and Banach algebras

Discussiones Mathematicae - General Algebra and Applications, 2014

Let R be a prime ring with extended centroid C, F a generalized derivation of R and n ≥ 1, m ≥ 1 fixed integers. In this paper we study the situations: 1. (F (x • y)) m = (x • y) n for all x, y ∈ I, where I is a nonzero ideal of R; 2. (F (x • y)) n = (x • y) n for all x, y ∈ I, where I is a nonzero right ideal of R. Moreover, we also investigate the situation in semiprime rings and Banach algebras.

Pair of Derivations on Semiprime Rings with Applications to Banach Algebras

Let R be an associative ring. An additive mapping d:R-→R is called a derivation if d(xy)=d(x)y+xd(y) holds for all x,y∈R. The objective of the present paper is to characterize a semiprime ring R which admits pair of derivations d and g such that [d(xm),g(yn)]=±[xm,y n ] for all x,y∈R or d(xm)∘g(yn)=±[x m ,y n ] for all x,y∈R or [d(x m ),d(y n )]=±g([x m ,y n ]) for all x,y∈R, where m and n are positive integers. With this, several results can be either deduced or generalized. Finally, we apply these purely algebraic results to obtain some range inclusion results of continuous derivations on Banach algebras.

Derivation in Prime Rings and Banach Algebra 1

2010

Let R be a prime ring of characteristics different from 2 and 3. If there exits a nonzero derivation d from R to itself that the map x → [[[[d(x), x], x], x], x] is centralizing on R then d = 0. Combining this result together withthe result of Sinclair and Johnson, we extend the Singer-Wermer theorem and its application in Banach algebra. Finally,we conclude some open problems. Mathematics Subject Classifications: 16A12, 16A70, 16W25, 16N60, 46K15

On Rings and Algebras with Derivations

Journal of Algebra and Its Applications, 2015

Let [Formula: see text] be an associative ring with center [Formula: see text] The objective of this paper is to discuss the commutativity of a semiprime ring [Formula: see text] which admits a derivation [Formula: see text] such that [Formula: see text] for all [Formula: see text] or [Formula: see text] for all [Formula: see text] or [Formula: see text] for all [Formula: see text] where [Formula: see text] and [Formula: see text] are fixed positive integers. Finally, we apply these purely ring theoretic results to obtain commutativity of Banach algebra via derivation.

ON DERIVATIONS SATISFYING CERTAIN IDENTITIES ON RINGS AND ALGEBRAS

ON DERIVATIONS SATISFYING CERTAIN IDENTITIES ON RINGS AND ALGEBRAS, 2018

The present paper deals with the commutativity of an associative ring R and a unital Banach Algebra A via derivations. Precisely, the study of multiplicative (generalized)-derivations F and G of semiprime (prime) ring R satisfying the identities G(xy) ± [F (x), y] ± [x, y] ∈ Z(R) and G(xy) ± [x, F (y)] ± [x, y] ∈ Z(R) has been carried out. Moreover, we prove that a unital prime Banach algebra A admitting continuous linear generalized derivations F and G is commutative if for any integer n > 1 either G((xy) n) + [F (x n), y n ] + [x n , y n ] ∈ Z(A) or G((xy) n) − [F (x n), y n ] − [x n , y n ] ∈ Z(A).

On generalized left derivations in rings and Banach algebras

Aequationes mathematicae, 2011

The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras. In fact, we prove the following results: Let R be a 2-torsion free semiprime ring, and let G : R −→ R be a generalized Jordan left derivation with associated Jordan left derivation δ : R −→ R. Then every generalized Jordan left derivation is a generalized left derivation on R. This result gives an affirmative answer to the question posed as a remark in Ashraf and Ali (Bull. Korean Math. Soc. 45:253-261, 2008). Also, the study of generalized left derivation has been made which acts as a homomorphism or as an anti-homomorphism on some appropriate subset of the ring R. Further, we introduce the notion of generalized left bi-derivation and prove that if a prime ring R admits a generalized left bi-derivation G with associated left bi-derivation B then either R is commutative or G is a right bi-centralizer(or bi-multiplier) on R. Finally, it is shown that every generalized Jordan left derivation on a semisimple Banach algebra is continuous.

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.