## Abstract

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 ringM_{3}.(Z) is equal to (ζ(2)^{2}ζ(3))^{-1}, where ζ is the Riemann zeta function.

Original language | English (US) |
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Pages (from-to) | 243-291 |

Number of pages | 49 |

Journal | Algebra and Number Theory |

Volume | 6 |

Issue number | 2 |

DOIs | |

State | Published - 2012 |

## Keywords

- Density
- Probability of generating
- Smallest number of generators

## ASJC Scopus subject areas

- Algebra and Number Theory