Characterization of Lie Derivations on Prime Rings

Abstract

Let ℛ be a unital prime ring with characteristic not 2 and containing a nontrivial idempotent P. It is shown that, under some mild conditions, an additive map L on ℛ satisfies L([A, B]) = [L(A), B] + [A, L(B)] whenever AB = 0 (resp., AB = P) if and only if it has the form L(A) = ϕ(A) + h(A) for all A ∈ ℛ, where ϕ is an additive derivation on ℛ and h is an additive map into its center.

Key Words:

• Derivations
• Lie derivations
• Prime rings

2000 Mathematics Subject Classification:

• 16W25
• 16W10
• 47B47

ACKNOWLEDGMENTS

The authors wish to give their thanks to the referees and the editor for their helpful comments and suggestions.

This work is partially supported by National Natural Science Foundation of China (No. 10771157, 10871111), Research Fund for the Doctoral Program of Higher Education of China (20101402110012), a joint Slovene-Chinese Grant Bl-CN/11-13/9-2, International Cooperation Project of Shanxi (2011081039), Tianyuan Funds of China (11026161) and Foundation of Shanxi University.

Notes

Communicated by M. Bresar.