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

Kravchenko, Rostyslav V.; Mazur, Marcin; Petrenko, Bogdan V.

doi:10.2140/ant.2012.6.243

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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

DOI: 10.2140/ant.2012.6.243

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 $M_{3} {(ℤ)}^{k}$ 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 $M_{3} (ℤ)$ is equal to ${(ζ {(2)}^{2} ζ (3))}^{- 1}$ , where $ζ$ 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

Milestones

Received: 9 May 2010

Revised: 8 January 2011

Accepted: 6 February 2011

Published: 24 June 2012

Authors

Rostyslav V. Kravchenko
Laboratoire de Mathématique d’Orsay Université Paris-Sud 91405 Orsay Cedex France

Marcin Mazur
Department of Mathematical Sciences Binghamton University Binghamton, NY 13902-6000 United States

Bogdan V. Petrenko
Department of Mathematics SUNY College at Brockport 350 New Campus Drive Brockport, NY 14420 United States

Vol. 6, No. 2, 2012