# Finite groups of units of commutative rings

**Luogo:** Dipartimento di Matematica, via Sommarive, 14 - Povo (TN) - Aula Seminari

**Ore** 11:30

**Relatore:**

**Roberto Dvornicich****(**Università di Pisa**)**

**Abstract: **

The aim of the talk (joint work with Ilaria Del Corso) is to answer the following question.

Assume that a ring A has a finite abelian group of units A∗: what are the possibilities for the structure of A∗? It is immdiately seen that one can reduce to the case of commutative rings.

Moreover, if a ring A has nilpotents, than the simple rule that a unit plus a nilpotent is againa unit roughly says that one can enlarge the group of units arbitrarily by just adding suitablenilpotents. There is however one simple but important constraint: the units must contain the units of the fundamental subring of A.

We shall show that A∗ can be any group of even finite order, but the family of possible groups of odd order is rather small.

Another interesting questions arises if one restricts to the case of reduced rings, i.e., without nilpotents. In this case, an important lemma allows to write the ring A as a direct product T × B, where T is the torsion subring T of A and B is a torsion-free reduced ring. We shall show that T∗ can be any group which is a product of multiplicative groups of finite fields, while B∗ can be almost any product of cyclic groups of order 2, 3 and 4, with a few exceptions that are completely classified.

**Referente: **Massimiliano Sala