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

Original · EN

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 ring M₃(Z) is equal to (ζ(2)² ζ(3))⁻¹, where ζ is the Riemann zeta-function.