# Algebra Seminar talk

2015-03-27

Kalle **Kaarli***On categorical equivalence of finite rings*

Abstract:

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(*p ^{m}*) and
GF(

*q*) with

^{n}*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?