## ABSTRACT

It has been known that given a Lie algebra of linear derivations of a free associative algebra, the subalgebra of constants is free (see [Jo78] and [Kh81]). A natural question that arises is to determine when this subalgebra is finitely generated. An analogous problem was considered for automorphisms. Invariants of

a free algebra under the action of a group of linear automorphisms form a free subalgebra (cf. [La76] and [Kh78]). In [DF82] and independently in [Kh84] it was proved that if G is a finite group of linear automorphisms of a free associative algebra of finite rank over a field, then the subalgebra of invariants is finitely generated if and only if the action of G is scalar. Somewhat later, W. Dicks gave a simplified proof of this fact. It is this proof that appears in [Co85, Theorem 10.4] and in [Pa89, Theorem 32.7]. The aim of this note is to show that Dicks’ proof can be adapted to

the case of derivations giving rise to the same conclusion. We shall then present a self-contained proof that the subalgebra of constants of a free algebra of finite rank over a field of positive characteristic under the action of a Lie algebra of linear derivations is finitely generated exactly when the derivations act scalarly. Our proof follows that of Dicks very closely for

the case of automorphisms with appropriate modifications where necessary. This result also follows from the work of A. Koryukin [Ko94].