On the group of units of a ring

L. Fuchs in [Abelian groups, 3rd edn (Pergamon, Oxford, 1960); Problem 72] posed the following problem: Characterize the groups which are the groups of all units in a commutative and associative ring with identity. A partial approach to this problem was suggested by Ditor in 1971, with the following less general question: Which whole numbers can be the number of units of a ring?

In the following years, these questions inspired the work of many authors, and some partial answer to them has been given. Among the others, we recall the work by Gilmer (1963), Hallett, Hirsch and Zassenhauss (1965-66), Pearson and Schneider (1970) , Dolzan (2002) and the recent papers by Chebolu and Lockridge (2015-17). Recently, in two joint papers with R. Dvornicich, we studied the original Fuchs’ question and we ”almost” answered it. In fact, we have been able to obtain a pretty good description of the possible groups of units equipped with families of examples of both realizable and non-realizable groups. We also examined the interesting case of torsion-free rings and we completely classified the possible finite abelian groups of units which arrise in this case. As a consequence of our results we completely answered Ditor’s question on the possible cardinalities of the group of units of a ring.