A Theorem on Derivations of Prime Rings with Involution

• Published in: Canadian journal of mathematics 1981-06, Vol.34 (2), p.356-369
• Main Author: Herstein, I. N.
• Format: Article
• Language: English
• ISSN: 0008-414X; 1496-4279
• DOI: 10.4153/CJM-1982-023-x

In a recent note [2] we showed that if R is a prime ring and d ≠ 0 a derivation of R such that d(x)d(y) = d(y)d(x) for all x, y ∈ R then, if R is not a characteristic 2, R must be commutative. (If char R = 2 we showed that R must be an order in a 4-dimensional simple algebra.) In this paper we shall consider a similar problem, namely, that of a prime ring R with involution * where d(x)d(y) = d(y)d(x) not for all x, y ∈ R but merely for symmetric elements x* = x and y* = y. Although it is clear that some results can be obtained if R is of characteristic 2, we shall only be concerned with the case char R ≠ 2. Even in this case one cannot hope to extend the result cited in the first paragraph, that is, to show that R is commutative.