Topology & Geometric Group Theory Seminar

Fall 2007

1:30 - 2:30, Malott 253

Tuesday, November 13

Any finite direct sum of matrix algebras (of size at least 2-by-2) over an infinite field has 2 generators. This is no longer true in general for a finite direct sum of matrix algebras over other commutative rings. We obtain an asymptotic upper bound for the smallest number of generators for a finite direct sum of matrix algebras over a finite field. We also obtain an exact formula for the smallest number a_n(q) of generators for a direct sum of n copies of the 2-by-2 matrix algebra over a field with q elements. We show a_n(2) is also the smallest number of generators for a direct sum of n copies of the 2-by-2 integer matrix ring. We remark that generators for a finite direct sum of integer matrix rings can be used as generators for a similar direct sum where the ring of integers is replaced with any ring with 1.

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).

The main motivation for this type of question comes from group theory, and the proof is based on group theory. The talk will include a discussion of related questions and known results in group theory.