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 M3()k admits two generators if and only if k 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 M3() is equal to (ζ(2)2ζ(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