Generalized Derivations with Engel Conditions on Polynomials

Abstract

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X ₁,…, X _t ) a nonzero polynomial in noncommutative indeterminates X ₁,…, X _t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X ₁,…, X _t ) are characterized if the Engel identity is satisfied: [D(f(x ₁,…, x _t )), f(x ₁,…, x _t )] _k  = 0 for all x ₁,…, x _t  ∈ R.

Key Words:

• Differential identity
• Engel condition
• Generalized derivation
• GPI
• Prime ring
• PI

2000 Mathematics Subject Classification:

• 16N60
• 16W25

ACKNOWLEDGMENT

The author would like to thank the referee for his suggestions and careful reading of the manuscript.

Notes

Communicated by M. Bresar.