The matrices below are assumed not to be 1-by-1. Any finite direct sum of matrix algebras with entries in an infinite field has 2 generators. This is no longer true in general for direct sum of matrix algebras with entries in other commutative rings. We obtain an asymptotic upper bound for the minimal number of generators for a finite direct sum of matrix algebras with entries in a finite field. This produces an upper bound for a similar quantity for integer matrix rings. We obtain an exact formula for the minimal number of generators for a finite direct sum of 2-by-2 matrix algebras with entries in a finite field. As a consequence, we show that a direct sum of up to 16 copies of M_2(Z) has 2 generators, i.e. every element of M_2(Z)^16 may be written as a noncommutative polynomial in these generators with integer coefficients. (Therefore, the same is true if in the previous sentence the ring of integers is replaced with any ring with 1.) It also follows that the minimal number of generators for the ring M_2(Z)^17 is 3. The talk will be based on my joint work with Said Sidki (Journal of Algebra, 310 (2007), no. 1, 15--40) and Rostyslav Kravchenko(arXiv:math/0611674).
R E F R E S H M E N T S
4:00 to 4:25 PM
Kenneth W. Anderson
Memorial Reading Room
