How Many Units Can a Commutative Ring Have?. Issue 10 (1st December 2017)

• Record Type: Journal Article
• Authors: Chebolu, Sunil K.; Lockridge, Keir
• Is Part Of: American Mathematical Monthly. Volume 124:Issue 10(2017)
• Page Start: 960
• Page End: 965
• Publication Date: 2017-12-01
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.4169/amer.math.monthly.124.10.960
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: László Fuchs posed the following problem in 1960, which remains open: determine whether a given abelian group can occur as the group of 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 units in a commutative ring. As a byproduct, we obtain a solution to Fuchs's problem for the class of finite abelian p -groups when p is an odd prime.