Sets of Elements that Pairwise Generate a Matrix Ring

In this article, we investigate the relationship between the minimum number of proper subgroups of GL(n, q) whose union is the whole GL(n, q) and the maximum number of elements that pairwise generate GL(n, q). We show that the minimum number of proper subrings of M _n (q) whose union is the whole M _n (q) is exactly the maximum number of elements that pairwise generate M _n (q).