Commuting Traces of Biadditive Maps Revisited

Abstract

Let R be a prime ring with char (R) ≠ 2. It is known that the form of a biadditive map B : R × R → R satisfying [B(x, x), x] = 0 for all x ∈ R can be described provided that R does not satisfy the standard polynomial identity S ₄. In this note we prove a linear algebraic result from which we are able to derive that the description of B is the same even when R satisfies S ₄. This makes it possible to complete the results on the structure of Lie isomorphisms and Lie derivations on prime rings.

Key Words:

• Commuting trace
• Biadditive map
• Prime ring
• Lie isomorphism
• Lie derivation

1991 Mathematics Subject Classifications:

• 16R50
• 15A03

Acknowledgments

The authors are thankful to Professor W.S. Martindale for suggesting the way to shorten the proof of Theorem 2.1. The authors were supported by the Ministry of Science of Slovenia.