On Constants of Algebraic Derivations and Fixed Points of Algebraic Automorphisms

Abstract

Let R be a semiprime algebra over a field F and d an algebraic derivation of R. We examine the relationship between R and the algebra of constants R d . We prove that: (1) The prime radical B ( R d ) is nilpotent with the index of nilpotency depending on the minimal polynomial of d; (2) R d is Artinian if and only if R is Artinian. Using these we obtain results about fixed subrings of algebraic automorphisms. For instance, we show that if σ is an automorphism of a prime order p of a semiprime ring R with pR = 0 then R is Artinian if and only if the fixed subring R σ is Artinian.