Skew Derivations and Engel Conditions

Abstract

It is known that for a nonzero derivation d of a prime ring R, if a nonzero ideal I of R satisfies the Engel-type identity [[…[[d(x ^{k ₀} ), x ^{k ₁} ], x ^{k ₂} ],…], x ^{k _n} ], then R is commutative. Here we extend this result to a skew derivation of R for a Lie ideal I, which has an immediate corollary that replaces d by an automorphism of R. A related result in two variables is obtained for d a (θ, ϕ)-derivation.

Key Words:

• Automorphism
• Differential identity
• Engel polynomial
• Prime ring
• Skew derivation

2010 Mathematics Subject Classification:

• Primary 16W25
• Secondary 16N60
• 16W10
• 16R50

Notes

Communicated by M. Bresar.