Power Values of Generalized Derivations with Annihilator Conditions in Prime Rings. Issue 7 (2nd July 2016)

• Record Type: Journal Article
• Authors: Ali, Asma; De Filippis, Vincenzo; Khan, Shahoor
• Is Part Of: Communications in algebra. Volume 44:Issue 7(2016)
• Page Start: 2887
• Page End: 2897
• Publication Date: 2016-07-02
• Subjects: Extended centroid -- Generalized derivation -- Prime ring -- Utumi quotient ring; 16N60 -- 16U80 -- 16W25; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2015.1065848
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract : Let R be a prime ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R, and F a generalized derivation with associated derivation d of R, m ≥ 1, n ≥ 1 two fixed integers, and 0 ≠ a ∈ R . Assume that a (( F ([ x, y ])) m − [ x, y ] n ) = 0, for all x, y ∈ I, then one of the following statements holds: R is commutative; n = m = 1 and there exists b ∈ Q such that F ( x ) = bx for all x ∈ R with ab = a ; There exists b ∈ C such that F ( x ) = bx for all x ∈ R with b m = 1 and [ x, y ] m = [ x, y ] n, for all x, y ∈ R ; R ⊆ M 2 ( C ), the ring of 2 × 2 matrices over C ; n = 1 and m ≥ 2 such that α m = α for all α ∈ C ; and there exists b ∈ Q such that F ( x ) = bx for all x ∈ R with ab = a ; R ⊆ M 2 ( C ) and char ( R ) = 2. Assume that char( R ) ≠ 2 and a (( F ([ x, y ])) m − [ x, y ] n ) ∈ Z ( R ) for all x, y ∈ I . If there exist x 0, y 0 ∈ I such that a (( F ([ x 0, y 0 ])) m − [ x 0, y 0 ] n ) ≠ 0, then either there exists a field E such that R ⊆ M 2 ( E ) or a ∈ Z ( R ), [ x, y ] m − [ x, y ] n ∈ Z ( R ) for any x, y ∈ R, and there exist b, c ∈ Z ( R ) sucht that ( b + c ) m = 1.