Derivations with invertible values in rings with involution

Abstract

Let R be a semiprime 2-torsion free ring with involution * and let S = { x e R I x = x*} be the set of symmetric elements. We prove that if R has a derivation d, non-zero on S, such that for all s e S either d(s) = 0 or d(s) is invertible, then R must be one of the following: (1) a division ring, (2) 2 X 2 matrices over a division ring, (3) the direct sum of a division ring and its opposite with exchange involution, (4) the direct sum of 2 X 2 matrices over a division ring and its opposite with exchange involution, (5) 4 X 4 matrices over a field with symplectic involution.