Actions of solvable Lie algebras on rings with no nilpotent elements
Let R be an algebra over a field K of characteristic \(p>0\) with no non- zero nilpotent elements, L be a finite dimensional solvable restricted Lie algebra over K and there is a Lie algebra homomorphism \(\phi: L\to Der_ K(R)\) which preserves pth powers. Let also \(R^ L=\{r\in R\); \(d(r)=0\), \(\delta\in \phi (L)\}\) be the ring of constants. The author proves the following main results: 1) if \(R^ L\) is a PI-algebra then R is also a PI-algebra; 2) R is left Goldie if and only if \(R^ L\) is left Goldie; 3) if R and \(R^ L\) are left Goldie then i) Q(R) is the localization of R at the regular elements of \(R^ L\); ii) \(Q(R^ L)=Q(R)^ L\) and \(R,R^ L\) have the same Goldie rank.
finite dimensional solvable restricted Lie algebra
scientific article; zbMATH DE number 4160963