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.

A recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has tr.A p k / Á tr.A p k 1 / .mod p k /. We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A; B are congruent modulo p k then the characteristic polynomials of A p and B p are congruent modulo p kC1 , and then we show that Arnold's conjecture follows from it easily. Using this result, we prove the following generalization of Euler's theorem for any 2 2 integral matrix A: the characteristic polynomials of Aˆ. n/ and Aˆ. n/ .n/ are congruent modulo n. Here is the Euler function, Q l iD1 p˛i i is a prime factorization of n andˆ.n/ D . .n/ C Q l iD1 p˛i 1 i .p i C 1//=2.

Abstract. We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras F G, where G is a finite group such that the order of G is invertible in F . We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2×2 matrices (A, B) and (A , B ) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M2(R), under an additional assumption that both pairs generate M2(R) as an R-algebra. Mathematics Subject Classification (2000). Primary 16H05, 16K20, 16S15, 16R30.

Let M n (Z) be the ring of n-by-n matrices with integral entries, and n 2. This paper studies the set G n (Z) of pairs (A, B) ∈ M n (Z) 2 generating M n (Z) as a ring. We use several presentations of M n (Z) with generators X = n i=1 E i+1,i and Y = E 11 to obtain the following consequences.(1) Let k 1. The following rings have presentations with 2 generators and finitely many relations: (a) k j =1 M m j (Q) for any m 1 , . . . , m k 2. (b) k j =1 M n j (Z), where n 1 , . . . , n k 2, and the same n i is repeated no more than three times.(2) Let D be a commutative domain of sufficiently large characteristic over which every finitely generated projective module is free. We use 4 relations for X and Y to describe all representations of the ring M n (D) into M m (D) for m n.(3) We obtain information about the asymptotic density of G n (F ) in M n (F ) 2 over different fields, and over the integers.

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