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RINGS WITH AN ELEMENTARY ABELIAN p-GROUP OF UNITS

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

JOURNAL OF COMMUTATIVE ALGEBRA（2023）

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Abstract
What are all rings R for which Rx (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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Key words
commutative rings,group of units,group algebras,Wedderburn-Artin,local rings
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