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

摘要

Let ? be a prime ring of characteristic different from 2, U{script} its right Ututmi quotient ring, C{script} its extended centroid, f(x _1,., x _n) a multilinear polynomial in n non-commuting variables over C{script} and S = { f(r _1,., r _n): r _1,., r _n ∈ ?}. Let F: ? → ? and G: ? → ? be non-zero generalized derivations on ?. We say that F and G are mutually strong Engel condition preserving (SEP for brevity) on S{script} if [G(x), F(y)]h = [x, y]h, for all x, y ∈ S{script} and fixed h ≥ 1. In this article we show that, if f(x _1,., x _n) is not central valued on ? and F, G are mutually SEP on S{script}, then one of the following holds: (a) there exists λ ∈ C{script} such that, for any x ∈ ?, G(x) = λx and F(x) = λ~(-h) x; (b) char(R) = p ≥ 3 and there exist λ ∈ C{script} and s ≥ 1 such that, for any x ∈ ?, G(x) = λx and is central valued on ?; (c) ? satisfies s _4, the standard identity of degree 4. The semiprime case for mutually SEP derivations on Lie ideals is also considered.