Commutativity of prime rings with symmetric generalized biderivations

Abstract

In this manuscript we have generalized the result of B. R. Reddy and C. J. S. Reddy [8]. We have taken R to be a prime ring with nonzero ideal I of R and G be a symmetric generalized biderivation of R . We have investigated the commutativity of R if G satisfies any one of the given identities (i) G ([ l,m ] ,n ) ± [ l,m ] = 0, (ii) G ( l ◦ m,n ) ± l ◦ m = 0, (iii) G ( l,m ) ◦ G ( m,n ) = 0, (iv) [ G ( l,m ) , G ( m,n )] = 0, (v) G ( l,m ) ◦ G ( m,n ) ± l ◦ n = 0 and (vi) [ G ( l,m ) , G ( m,n )] ± [ l,n ] = 0 ∀ l,m,n ∈ I .