Derivations with invertible values in rings with involution

Antonio Giambruno, P. Misso and Francisco César Polcino Milies

Vol. 123 (1986), No. 1, 47–54

DOI: 10.2140/pjm.1986.123.47

Abstract

Let R be a semiprime 2-torsion free ring with involution ^∗ and let S = {x ∈ R|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 ∈ S either d(s) = 0 or d(s) is invertible, then R must be one of the following: (1) a division ring, (2) 2 × 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 × 2 matrices over a division ring and its opposite with exchange involution, (5) 4 × 4 matrices over a field with symplectic involution.

Mathematical Subject Classification

Primary: 16A72, 16A72

Secondary: 16A28

Milestones

Received: 6 March 1984

Revised: 5 July 1984

Published: 1 May 1986