In this paper, we describe the behavior of Lie n-centralizers acting on the specific subsets within unital prime rings of characteristic not 2. Subsequently, we employ these findings to characterize generalized Lie n-derivations on such rings. Our results extend previous research on standard operator algebras on Banach spaces and von Neumann factors. Moreover, we introduce characterizations for generalized Lie nderivations at local products when restricted to factor von Neumann algebras. As an application of the obtained results, we present some variants of Posner’s second theorem on prime rings.
