Differential identities in prime rings with involution

Abstract:

Let $R$ be a prime ring with involution. Using work of V. K. Kharchenko it is shown that any generalized identity for $R$ involving derivations of $R$ and the involution of $R$ is a consequence of the generalized identities with involution which $R$ satisfies. In obtaining this result, a generalization, to rings satisfying a GPI, of the classical theorem characterizing inner derivations of finite-dimensional simple algebras is required. Consequences of the main theorem are that in characteristic zero no outer derivation of $R$ can act algebraically on the set of symmetric elements of $R$, and if the images of the set of symmetric elements under the derivations of $R$ satisfy a polynomial relation, then $R$ must satisfy a generalized polynomial identity.