Rostyslav V. Kravchenko

arXiv2014

Generators of maximal orders

Let R be the ring of algebraic integers in a number field K and let L be a maximal order in a semisimple K-algebra B. Building on our previous work, we compute the smallest number of algebra generators of L considered as an R-algebra. This reproves and vastly extends the results of P.A.B. Pleasants, who considered the case when B is a number field. In order to achieve our goal, we obtain several results about counting generators of algebras which have finitely many elements. These results should be of independent interest.

math.RA↗

arXiv2010

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

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 $M_3(\mathbb{Z})^{k}$ admits two generators if and only if $k\leq 768$. 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 $3\times 3$ matrices generate the ring $M_3(\mathbb{Z})$ is equal to $(ζ(2)^2 ζ(3))^{-1}$, where $ζ$ is the Riemann zeta-function.

math.RA↗