Let \(\mathfrak {R}\) be any non-commutative prime ring of char \(\mathfrak {R}\ne 2\) , \(\mathfrak {U}\) its Utumi quotient ring, \(\mathfrak {L}\) a non-central Lie ideal of \(\mathfrak {R}\) and \(\mathfrak {F}\) , \(\mathfrak {G}\) two non-zero b-generalized derivations of \(\mathfrak {R}\) . Suppose that \([\mathfrak {F}(u^{n_{1}})u^{n_{2}}-u^{n_{3}}\mathfrak {G}(u^{n_{4}}), u^{n_{5}}]_{n}=0\) where u varies over \( \mathfrak {L}\) , fixed \(n,n_{1},n_{2},n_{3},n_{4},n_{5}\in \mathbb {Z}^{+}\) . We describe all possibilities for the forms of \(\mathfrak {F}\) and \(\mathfrak {G}\) and also give some affirmations about the structure of \(\mathfrak {R}.\) Consequently, we extend the results of Liu [20] and of Khan et al. [14].
