On pairs of matrices generating matrix rings and their presentations

Let be the ring of n-by-n matrices with integral entries, and n 2. This paper studies the set of pairs generating as a ring. We use several presentations of with generators and Y=E11 to obtain the following consequences. (1) Let k 1. The following rings have presentations with 2 generators and finitely many relations: (a) for any m1,…,mk 2. (b) , where n1,…,nk 2, and the same ni 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 Mn(D) into Mm(D) for m n. (3) We obtain information about the asymptotic density of Gn(F) in Mn(F)2 over different fields, and over the integers.