PI-Algebras Generated by Nilpotent Elements of Bounded Index

✍ Scribed by David M. Riley

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 191 KB
• Volume: 192
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.6940

✦ Synopsis

Let R be an associative algebra over a field F of positive characteristic p. We address the following problem: if R is generated by nilpotent elements with bounded index, under what conditions can we conclude that R itself is nil of bounded index? We prove that whenever R satisfies the Engel condition and p ) 0, or R is Lie soluble and p ) 2, then the conclusion holds. Along the way we prove that if R satisfies the Engel condition, then there exists a positive integer n such that the map x ¬ x p n is additive. The converse also holds when F is infinite or R is generated by nilpotent elements.