Results on b-Generalized Derivations in Rings

This paper studies various identities of noncommutative prime rings involving b-generalized derivations. In particular, we prove that if a noncommutative prime ring \(\mathfrak {R}\) admits a nonzero b-generalized derivation \(\mathfrak {F}\) such that \([\mathfrak {F}(l), m]=[l, \mathfrak {F}(m)]\) for all \(l,m\in \mathfrak {I}\) , where \(\mathfrak {I}\) is a nonzero ideal of \(\mathfrak {R}\) , then \(\mathfrak {F}(l)=a l\) for all \(l\in \mathfrak {R}\) and for some \(a\in \mathcal {C}\) unless \(\mathfrak {R}\subseteq M_2(F)\) , the \(2\times 2\) matrix ring over the field F. This gives a natural generalization of the results for derivations, generalized derivations and generalized \(\rho \) -derivations with an X-inner automorphism.