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 → ∞.