On the smallest number of generators and the probability of generating an algebra

TL;DR

This paper investigates the minimal number of generators needed for algebras over number field orders and computes the probability that random elements generate such algebras, providing explicit formulas and asymptotic results.

Contribution

It introduces a method to determine the smallest number of generators for algebras over number field orders and derives probabilistic formulas for generating sets.

Findings

The ring M_3(Z)^k is 2-generated iff k ≤ 768.

Probability that two random 3x3 matrices generate M_3(Z) is (ζ(2)^2 ζ(3))^{-1}.

Develops a technique to compute minimal generators and generating probabilities.

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 $(\zeta(2)^2 \zeta(3))^{-1}$, where $\zeta$ is the Riemann zeta-function.