On pairs of matrices generating matrix rings and their presentations

Let Mn(Z) be the ring of n-by-n matrices with integral entries, and n⩾2. This paper studies the set Gn(Z) of pairs (A,B)∈Mn2(Z) generating Mn(Z) as a ring. We use several presentations of Mn(Z) 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 Mn2(F) over different fields, and over the integers.