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

Abstract

On the smallest number of generators and the probability of generating an algebra Rostyslav V. Kravchenko, Marcin Mazur and Bogdan V. PetrenkoIn 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 ‫)ޚ(‬ 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 M 3 ‫)ޚ(‬ is equal to (ζ (2) 2 ζ (3)) −1 , where ζ is the Riemann zeta function.

Generators of maximal orders

“…Any such an algebra is of the form M n (F ), where F is a finite field. In this section we extend some results about generators of matrix rings over finite fields obtained in [3]. Let us first recall some of the results from [3].…”

“…Indeed, as we will see in the course of the proof of Theorem 2.2, F is a field and each quotient A i /J(A i ) is a finite dimensional simple F -algebra. If J(A) = 0, then J(A i ) = 0 for all i and A can be considered as a product of m copies of a finite dimensional simple F -algebra as in the Theorem 6.1 of [3].…”

“…We start this section by recalling a few results from [3]. For a commutative ring R and an R-algebra A which is finitely generated as an R-module we write r(A, R) for the smallest number k such that A can be generated by k elements as an R-algebra.…”

“…Suppose that A is a free R-module of finite rank. In [3] we introduced the notion of density den k (A) of the set of all k-tuples in A k which generate A as and R-algebra. Roughly speaking, a choice of an integral basis of R and of a basis of A over R allows us to introduce integral coordinates on all cartesian powers A k , k ∈ N. For any subset S of A k and any N we consider the finite set S(N) of all points whose coordinates are in the interval [−N, N].…”

Random Nilpotent Groups I

“…Dunfield and Thurston [5] use a lazy random walk: ℓ letters are chosen uniformly from the (2m+1) possibilities of a ± i and the identity letter, creating a word of length ≤ ℓ, whose abelianization becomes a column of M . Results by Kravchenko-Mazur-Petrenko [9] and Wang-Stanley [17] use the standard "box" model: integer entries are drawn uniformly at random from [−ℓ, ℓ], and asymptotics are calculated as ℓ → ∞. (This is the most classical way to randomize integers in number theory; see [8].)…”