Mathematics – Rings and Algebras
Scientific paper
2010-01-17
Mathematics
Rings and Algebras
58 pages; a minor typo was corrected in formula (14)
Scientific paper
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 $(\zeta(2)^2 \zeta(3))^{-1}$, where $\zeta$ is the Riemann zeta-function.
Kravchenko Rostyslav V.
Mazur Marcin
Petrenko Bogdan V.
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