PI-Algebras Generated by Nilpotent Elements of Bounded Index Original Research Article

LetRbe an associative algebra over a fieldFof positive characteristicp. We address the following problem: ifRis generated by nilpotent elements with bounded index, under what conditions can we conclude thatRitself is nil of bounded index? We prove that wheneverRsatisfies the Engel condition andp > 0, orRis Lie soluble andp > 2, then the conclusion holds. Along the way we prove that ifRsatisfies the Engel condition, then there exists a positive integernsuch that the mapx maps to xpnis additive. The converse also holds whenFis infinite orRis generated by nilpotent elements.