#### Vol. 6, No. 2, 2012

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

### Rostyslav V. Kravchenko, Marcin Mazur and Bogdan V. Petrenko

Vol. 6 (2012), No. 2, 243–291
##### 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 ${M}_{3}{\left(ℤ\right)}^{k}$ admits two generators if and only if $k\le 768$. 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 $3×3$ matrices generate the ring ${M}_{3}\left(ℤ\right)$ is equal to ${\left(\zeta {\left(2\right)}^{2}\zeta \left(3\right)\right)}^{-1}$, where $\zeta$ is the Riemann zeta function.

##### Keywords
density, smallest number of generators, probability of generating
##### Mathematical Subject Classification 2000
Primary: 16S15, 11R45, 11R99, 15A33, 15B36, 11C20, 11C08
Secondary: 16P10, 16H05