Algebra Seminar talk

Kalle Kaarli
On categorical equivalence of finite rings

This is a joint work with Oleg Koshik (Tartu) and Tamás Waldhauser (Szeged)

Two (universal) algebras A and B are called categorically equivalent if there is a categorical equivalence between the varieties they generate that maps A to B. Here we consider categorical equivalence in the variety of (associative) rings with unity element. It is known that the Galois fields GF(pm) and GF(qn) with p and q primes are categorically equivalent if and only if m=n. Our aim was to find new, essentially different examples and to try to solve the problem in general, that is, to describe all pairs of finite categorically equivalent rings.

The main results of the present work are the following.

1. The general problem of categorical equivalence between finite rings was reduced to the case of rings of prime power characteristics.

2. It was proved that a ring categorically equivalent to a finite semisimple ring is finite semisimple, too.

3. The problem when are two finite semisimple rings categorically equivalent, was completely solved.

4. It was proved that finite categorically equivalent rings of coprime characteristics must be semisimple. 

Problem. Do there exist finite non-semisimple rings that are

categorically equivalent but are neither isomorphic nor dually isomorphic?