Strong commutativity and Engel condition preserving maps in prime and semiprime rings

Abstract

Let ℛ be a prime ring of characteristic different from 2, 𝒰 its right Ututmi quotient ring, 𝒞 its extended centroid, f(x 1, … , x n ) a multilinear polynomial in n non-commuting variables over 𝒞 and S = { f(r 1, … , r n ) : r 1, … , r n ∈ ℛ}. Let F: ℛ → ℛ and G: ℛ → ℛ be non-zero generalized derivations on ℛ. We say that F and G are mutually strong Engel condition preserving (SEP for brevity) on 𝒮 if [G(x), F(y)] h = [x, y] h , for all x, y ∈ 𝒮 and fixed h ≥ 1. In this article we show that, if f(x 1, … , x n ) is not central valued on ℛ and F, G are mutually SEP on 𝒮, then one of the following holds: (a) there exists λ ∈ 𝒞 such that, for any x ∈ ℛ, G(x) = λx and F(x) = λ−h x; (b) char(R) = p ≥ 3 and there exist λ ∈ 𝒞 and s ≥ 1 such that, for any x ∈ ℛ, G(x) = λx and is central valued on ℛ; (c) ℛ satisfies s 4, the standard identity of degree 4. The semiprime case for mutually SEP derivations on Lie ideals is also considered.