One-Sided Ideal Growth of Free Associative Algebras

Abstract

Let R be a finitely generated associative algebra with unity over a finite field \({\Bbb F}_q\). Denote by a n (R) the number of left ideals J ⊂ R such that dim R/J = n for all n ≥ 1. We explicitly compute and find asymptotics of the left ideal growth for the free associative algebra A d of rank d with unity over \({\Bbb F}_q\), where d ≥ 1. This function yields a bound a n (R) ≤ a n (A d ), \(n\in{\Bbb N}\), where R is an arbitrary algebra generated by d elements. Denote by m n (R) the number of maximal left ideals J ⊂ R such that dim R/J = n, for n ≥ 1. Let d ≥ 2, we prove that m n (A d ) ≈ a n (A d ) as n → ∞.