On the smallest number of generators and the probability of generating an algebra.
Abstract
In this paper we study algebraic and asymptotic properties of generating sets
of algebras over orders in number fields. Let $A$ be an associative algebra
over an order $R$ in an algebraic number field. We assume that $A$ is a free
$R$-module of finite rank. We develop a technique to compute the smallest
number of generators of $A$. For example, we prove that the ring
$\M_3(\mathbb{Z})^{k}$ admits two generators if and only if $k\leq 768$. For a
given positive integer $m$, we define the density of the set of all ordered
$m$-tuples of elements of $A$ which generate it as an $R$-algebra. We express
this density as a certain infinite product over the maximal ideals of $R$, and
we interpret the resulting formula probabilistically. For example, we show that
the probability that 2 random $3\times 3$ matrices generate the ring
$\M_3(\mathbb{Z})$ is equal to $\dd (\zeta(2)^2 \zeta(3))^{-1}$, where $\zeta$
is the Riemann zeta-function.
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