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.