TY - JOUR

T1 - On the smallest number of generators and the probability of generating an algebra

AU - Kravchenko, Rostyslav V.

AU - Mazur, Marcin

AU - Petrenko, Bogdan V.

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

KW - Density

KW - Probability of generating

KW - Smallest number of generators

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U2 - 10.2140/ant.2012.6.243

DO - 10.2140/ant.2012.6.243

M3 - Article

AN - SCOPUS:84863430551

VL - 6

SP - 243

EP - 291

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 2

ER -