A note on derivations of commutative algebras

Abstract

It is well known that the (solvable) radical of a Lie or Jordan algebra is invariant under all derivations of the algebra if the groundfield is not modular [4] and [5]. In this note we obtain a similar result for commutative power-associative algebras of degree one by following Jacobson's argument in [5] and then appealing to a theorem of Gerstenhaber [3] at that point where the Jordan identity was required. Our result (Theorem 1) seems useful in classifying simple algebras of degree one which satisfy identities giving rise to derivations of the algebras. For example, an immediate corollary to Theorem 1 is Kleinfeld and Kokoris' determination of simple flexible algebras of degree one [6]. Then in Theorem 2 we characterize the simple degree one algebras which satisfy identities considered by Kosier [7], Osborn [8], and the author [2]. Before giving Theorem 1, it is necessary to state some definitions and elementary identities. In an algebra A the associator (x, y, z) and commutator (x, y) are defined for each x, y, z in A by the equations (x, y, z) = (xy)z-x(yz) and (x, y) =xy-yx. The following identity may easily be verified.