Two-Sided Annihilator Condition with Generalized Derivations on Multilinear Polynomials

• Published in: International scholarly research notices 2014, Vol.2014, p.563284-9
• Main Authors: De Filippis, V., Scudo, G., Sorrenti, L.
• Format: Article
• Language: English
• Subjects: Algebra; Mathematical research; Polynomials
• ISSN: 2356-7872; 2356-7872
• DOI: 10.1155/2014/563284

Let R be a prime ring of characteristic different from 2, with extended centroid C , U its two-sided Utumi quotient ring, F a nonzero generalized derivation of R , f ( x 1 , … , x n ) a noncentral multilinear polynomial over C in n noncommuting variables, and a , b ∈ R such that a [ F ( f ( r 1 , … , r n ) ) , f ( r 1 , … , r n ) ] b = 0 for any r 1 , … , r n ∈ R . Then one of the following holds: (1) a = 0 ; (2) b = 0 ; (3) there exists λ ∈ C such that F ( x ) = λ x , for all x ∈ R ; (4) there exist q ∈ U and λ ∈ C such that F ( x ) = ( q + λ ) x + x q , for all x ∈ R , and f ( x 1 , … , x n ) 2 is central valued on R ; (5) there exist q ∈ U and λ , μ ∈ C such that F ( x ) = ( q + λ ) x + x q , for all x ∈ R , and a q = μ a , q b = μ b .