RINGS WITH AN ELEMENTARY ABELIAN p-GROUP OF UNITS

Sunil K. Chebolu; Jeremy Corry; Elizabeth Grimm; Andrew Hatfield

doi:10.1216/jca.2023.15.469

Abstract

What are all rings $R$ for which $R^\times$ (the group of invertible elements of $R$ under multiplication) is an elementary abelian $p$ -group? We answer this question for finite-dimensional commutative $k$ -algebras, finite commutative rings, modular group algebras, and path algebras. Two interesting byproducts of this work are a characterization of Mersenne primes and a connection to Dedekind’s problem.

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Sunil K. Chebolu. Jeremy Corry. Elizabeth Grimm. Andrew Hatfield. "RINGS WITH AN ELEMENTARY ABELIAN $\p$ -GROUP OF UNITS." J. Commut. Algebra 15 (4) 469 - 480, Winter 2023. https://doi.org/10.1216/jca.2023.15.469

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Received: 6 December 2021; Revised: 7 September 2022; Accepted: 16 December 2022; Published: Winter 2023

First available in Project Euclid: 20 December 2023

MathSciNet: MR4680631

Digital Object Identifier: 10.1216/jca.2023.15.469

Subjects:

Primary: 11T06

Secondary: 16U60

Keywords: commutative rings , group algebras , group of units , local rings , Wedderburn–Artin

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