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.

Keywords:

• generalized polynomial identity
• skew derivation
• prime and semiprime ring

AMS Subject Classifications::

• 16R50
• 16W25
• 16N60

Acknowledgements

The authors would like to thank the referee for her/his suggestions and careful reading of this article.