An identity on automorphisms of Lie ideals in prime rings

Abstract

In the present paper it is shown that a prime ring \(R\) with center \(Z\) satisfies \(s_4\), the standard identity in four variables if \(R\) admits a non-identity automorphism \(\sigma\) such that \( [u, v] - u^{m}[u^\sigma,u]^nu^\sigma\in Z\) for all \(u\) in some noncentral ideal \(L\) of \(R\), whenever \(\operatorname{char}(R)>n+m\) or \(\operatorname{char}(R)=0\), where \(n\)  and \(m\) are fixed positive integer.