# How many units can a commutative ring have?

## TL;DR

This paper classifies the possible sizes of the unit groups in commutative rings, providing an elementary solution to a simplified problem and addressing a special case of Fuchs' longstanding open question.

## Contribution

It offers a new elementary approach to determine all possible cardinalities of unit groups in commutative rings, including finite abelian p-groups for odd primes.

## Key findings

- Characterization of all possible unit group cardinalities
- Solution for finite abelian p-groups with odd primes
- Addresses a simplified version of Fuchs' problem

## Abstract

Laszlo Fuchs posed the following problem in 1960, which remains open: classify the abelian groups occurring as the group of all units in a commutative ring. In this note, we provide an elementary solution to a simpler, related problem: find all cardinal numbers occurring as the cardinality of the group of all units in a commutative ring. As a by-product, we obtain a solution to Fuchs' problem for the class of finite abelian $p$-groups when $p$ is an odd prime.