## Topology & Geometric Group Theory Seminar

## Fall 2007

### 1:30 - 2:30, Malott 253

Tuesday, November 13

**Bogdan
Petrenko**, SUNY Brockport

*Some formulas for the smallest number of generators for finite
direct sums of matrix algebras
*

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.

Back to seminar home page.