Affiliation:
1. Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.
2. Einstein Institute of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel
Abstract
Let [Formula: see text] be a residually finite dimensional algebra (not necessarily associative) over a field [Formula: see text]. Suppose first that [Formula: see text] is algebraically closed. We show that if [Formula: see text] satisfies a homogeneous almost identity [Formula: see text], then [Formula: see text] has an ideal of finite codimension satisfying the identity [Formula: see text]. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra [Formula: see text] over [Formula: see text] is almost [Formula: see text]-Engel, then [Formula: see text] has a nilpotent (respectively, locally nilpotent) ideal of finite codimension if char [Formula: see text] (respectively, char [Formula: see text]). Next, suppose that [Formula: see text] is finite (so [Formula: see text] is residually finite). We prove that, if [Formula: see text] satisfies a homogeneous probabilistic identity [Formula: see text], then [Formula: see text] is a coset identity of [Formula: see text]. Moreover, if [Formula: see text] is multilinear, then [Formula: see text] is an identity of some finite index ideal of [Formula: see text]. Along the way we show that if [Formula: see text] has degree [Formula: see text], and [Formula: see text] is a finite [Formula: see text]-algebra such that the probability that [Formula: see text] (where [Formula: see text] are randomly chosen) is at least [Formula: see text], then [Formula: see text] is an identity of [Formula: see text]. This solves a ring-theoretic analogue of a (still open) group-theoretic problem posed by Dixon,
Funder
National Science Foundation
United States - Israel Binational Science Foundation
Israel Science Foundation
Publisher
World Scientific Pub Co Pte Lt
Subject
Applied Mathematics,Algebra and Number Theory
Reference9 articles.
1. [B] Bourbaki, Nicolas: Éléments de mathématique. Algèbre. Chapitres 4 à 7. Masson, Paris, 1981.
2. [DF] Drensky, Vesselin; Formanek, Edward: Polynomial identity rings. Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2004.
3. [LS] Larsen, Michael; Shalev, Aner: Almost PI algebras are PI, arXiv:1910.05764, to appear Proc. Amer. Math. Soc.