Displaying similar documents to “An identity with generalized derivations on Lie ideals, right ideals and Banach algebras”

Derivations with Engel conditions in prime and semiprime rings

Shuliang Huang (2011)

Czechoslovak Mathematical Journal

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Let R be a prime ring, I a nonzero ideal of R , d a derivation of R and m , n fixed positive integers. (i) If ( d [ x , y ] ) m = [ x , y ] n for all x , y ∈ I , then R is commutative. (ii) If Char R ≠ 2 and [ d ( x ) , d ( y ) ] m = [ x , y ] n for all x , y ∈ I , then R is commutative. Moreover, we also examine the case when R is a semiprime ring.

Derivations with power central values on Lie ideals in prime rings

Basudeb Dhara, Rajendra K. Sharma (2008)

Czechoslovak Mathematical Journal

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Let R be a prime ring of char R ≠ 2 with a nonzero derivation d and let U be its noncentral Lie ideal. If for some fixed integers n 1 ≥ 0 , n 2 ≥ 0 , n 3 ≥ 0 , ( u n 1 [ d ( u ) , u ] u n 2 ) n 3 ∈ Z ( R ) for all u ∈ U , then R satisfies S 4 , the standard identity in four variables.

Generalized derivations in prime rings and Banach algebras

Asma Ali, Basudeb Dhara, Shahoor Khan (2014)

Discussiones Mathematicae - General Algebra and Applications

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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.

On left ( θ , ϕ ) -derivations of prime rings

Mohammad Ashraf (2005)

Archivum Mathematicum

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Let R be a 2 -torsion free prime ring. Suppose that θ , φ are automorphisms of R . In the present paper it is established that if R admits a nonzero Jordan left ( θ , θ ) -derivation, then R is commutative. Further, as an application of this resul it is shown that every Jordan left ( θ , θ ) -derivation on R is a left ( θ , θ ) -derivation on R . Finally, in case of an arbitrary prime ring it is proved that if R admits a left ( θ , φ ) -derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal...

Posner's second theorem and annihilator conditions with generalized skew derivations

Vincenzo De Filippis, Feng Wei (2012)

Colloquium Mathematicae

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Let be a prime ring of characteristic different from 2, r be its right Martindale quotient ring and be its extended centroid. Suppose that is a non-zero generalized skew derivation of and f(x₁,..., xₙ) is a non-central multilinear polynomial over with n non-commuting variables. If there exists a non-zero element a of such that a[ (f(r₁,..., rₙ)),f(r₁, ..., rₙ)] = 0 for all r₁, ..., rₙ ∈ , then one of the following holds: (a) there exists λ ∈ such that (x) = λx for all x ∈ ; (b) there...

On Jordan ideals and derivations in rings with involution

Lahcen Oukhtite (2010)

Commentationes Mathematicae Universitatis Carolinae

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Let R be a 2 -torsion free * -prime ring, d a derivation which commutes with * and J a * -Jordan ideal and a subring of R . In this paper, it is shown that if either d acts as a homomorphism or as an anti-homomorphism on J , then d = 0 or J ⊆ Z ( R ) . Furthermore, an example is given to demonstrate that the * -primeness hypothesis is not superfluous.