(untitled) (JOURNAL OF ADVANCES IN MATHEMATICS, 2015)
Let R be a prime ring of characteristic dierent from 2, L a non-central Lie ideal of R, and m; n xed positive integers. If R admits a generalized derivation F associated with a deviation d such that F(u)2)m -(F(u))2n ÏµZ(R) for all u Ïµ L, then R satises S4 , the standard identity in four variables. Moreover, we also examine the case when R is semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrallybounded generalized derivations on Banach algebras.