On Lie ideals and $ (\sigma, \tau) $-Jordan derivations on prime rings

Abstract

Let $ R $ be a prime ring with characteristic different from two and let $ U $ be a Lie ideal of $ R $ such that $ u^2 \in U $ for all $ u \in U $. Suppose that $ \sigma, \tau $ are automorphisms of $ R $. In the present paper, it is shown that if $ d $ is an additive mapping of $ R $ into itself satisfying $ d (u^2) = d(u) \sigma (v) + \tau (u) d(v) $ for all $ u,v \in U $, then $ d(uv) = d(u) \sigma (v) + \tau(u) d(v) $ for all $ u, v \in U $.