There has been considerable interest in recent decades in questions of random generation of finite and profinite groups and finite simple groups in particular. In this paper, we study similar notions for finite and profinite associative algebras. Let $k={\mathbb{F}}_q$ be a finite field. Let $A$ be a finite-dimensional, associative, unital algebra over $k$. Let $P(A)$ be the probability that two elements of $A$ chosen (uniformly and independently) at random will generate $A$ as a unital $k$-algebra. It is known that if $A$ is simple, then $P(A)\to 1$ as $|A|\to \infty$. We extend this result to a large class of finite associative algebras. For $A$ simple, we find the optimal lower bound for $P(A)$ and we estimate the growth rate of $P(A)$ in terms of the minimal index $m(A)$ of any proper subalgebra of $A$. We also study the random generation of simple algebras $A$ by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of general finite algebras. Finally, we let $A$ be a profinite algebra over $k$. We show that $A$ is positively finitely generated if and only if $A$ has polynomial maximal subalgebra growth. Related quantitative results are also established.

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随机生成关联代数

近几十年来，人们对有限群和有限群，特别是有限单群的随机生成问题产生了相当大的兴趣。在本文中，我们研究有限和有限关联代数的类似概念。让$k={\mathbb{F}}_q$是一个有限域。让$A$是一个有限维、结合、单位代数$k$。让$磷（A）$是两个元素的概率$A$随机选择（统一且独立）将生成$A$作为一个整体$k$-代数。据了解，如果$A$很简单，那么$磷（A）\to 1$作为$|A|\to \mathrm{无穷大}$。我们将这个结果扩展到一大类有限关联代数。为了$A$简单，我们找到最佳下界$磷（A）$我们估计增长率$磷（A）$就最小索引而言$米（A）$的任何真子代数$A$。我们还研究简单代数的随机生成$A$由具有给定特征多项式（或给定秩）的两个元素组成。此外，我们将一般有限代数的生成元的最小数量限制在上面和下面。最后，我们让$A$是一个有余代数$k$。我们表明$A$是正有限生成当且仅当$A$具有多项式最大子代数增长。还建立了相关的定量结果。