Actions of solvable Lie algebras on rings with no nilpotent elements

Abstract

Let R be an algebra with no non-zero nilpotent elements acted on by a finite dimensional solvable restricted Lie algebra L. We examine the relationship between R and the ring of constants R L . In particular, we prove: 1. (1) if R L satisfies a polynomial identity of degree d, then R satisfies a polynomial identity of degree p n d, where n is the dimension of L; 2. (2) R is Goldie if and only if R L is Goldie; 3. (3) in the case where both R and R L are Goldie, both R and R L have the same Goldie rank and the Goldie localization of R can be obtained by inverting the regular elements of R L .