On Commutativity of Rings With Derivations

Ashraf, Mohammad; Rehman, Nadeem ur

doi:10.1007/bf03323547

Cited by 126 publications

(63 citation statements)

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“…Recently, a considerable number of researchers have investigated the ideals in prime rings as well as the commutativity of prime rings that consider derivations and generalized derivations, see for example [2], [3], [5] and [7]. In [4], Ashraf and Khan showed that a * -ideal U is central if the ring R admits a general derivation F associated with a derivation d satisfying specific properties.…”

Section: Introductionmentioning

confidence: 99%

On Ideals and Commutativity of Prime Rings with Generalized Derivations

Nawas

AL-Omary²

2018

Eur. J. Pure Appl. Math.

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Abstract. 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) = xF (y) + d(x)y holds for all x, y ∈ R. It is called a generalized (α, β)−derivation on R if there exists an (α, β)−derivation d: R → R such that the equation F (xy) = F (x)α(y) + β(x)d(y) holds for all x, y ∈ R. In the present paper, we investigate commutativity of a prime ring R, which satisfies certain differential identities on the left ideals of R. Moreover some results on commutativity of rings with involutions that satisfy certain identities are proved.

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Section: Introductionmentioning

confidence: 99%

On Ideals and Commutativity of Prime Rings with Generalized Derivations

Nawas

AL-Omary²

2018

Eur. J. Pure Appl. Math.

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“…In 2002, Ashraf and Rehman [3] discussed the action of derivation in prime ring. More precisely they showed that if R is a 2-torsion free prime ring, I is a nonzero ideal of R and d is a nonzero derivation of R such that d(x) • d(y) = x • y for all x, y ∈ I, then R is commutative.…”

Section: Introductionmentioning

confidence: 99%

On generalized derivation in rings and Banach algebras

Raza¹,

Rehman²

2017

Kragujevac J Math

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Abstract. Let R be a prime ring, F be a generalized derivation associated with a derivation d of R and m, n be the fixed positive integers. In this paper we study the case when one of the following holds: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 obtain some range inclusion results of continuous or spectrally bounded generalized derivations on non-commutative Banach algebras.

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“…This paper is included in a line of investigation concerning the relationship between the structure of a ring R and the behaviour of some additive mappings defined on R satisfy certain special identities. In [1], Ashraf and Rehman proved that if R is a prime ring, I a nonzero ideal of R and d is a derivation of R such that d(x • y) = x • y for all x, y ∈ I, then R is commutative. In Proof: Since R is a prime ring and F is a generalized derivation of R, by Lee [13,Theorem 3], F (x) = ax + d(x) for some a ∈ U and a derivation d on U .…”

Section: Introductionmentioning

confidence: 99%