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

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

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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.(Z) 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 ringM 3.(Z) is equal to (ζ(2) 2ζ(3)) -1, where ζ is the Riemann zeta function.

Original languageEnglish (US)
Pages (from-to)243-291
Number of pages49
JournalAlgebra and Number Theory
Volume6
Issue number2
DOIs
StatePublished - 2012

Keywords

  • Density
  • Probability of generating
  • Smallest number of generators

ASJC Scopus subject areas

  • Algebra and Number Theory

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