Abstract: | Let R be a finitely generated associative algebra with unity over a finite field
. 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
, where d ≥ 1. This function yields a bound a
n
(R) ≤ a
n
(A
d
),
, 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 → ∞. |