Rings with an elementary abelian $p$-group of units

TL;DR

This paper characterizes rings whose units form an elementary abelian p-group, exploring various algebraic structures and revealing connections to Mersenne primes and Dedekind's problem.

Contribution

It provides a comprehensive classification of such rings across multiple algebraic contexts, linking group-theoretic properties to number theory and combinatorics.

Findings

Characterization of rings with elementary abelian p-group of units

Connection between units in rings and Mersenne primes

Link to Dedekind's problem in algebraic structures

Abstract

What are all rings $R$ for which $R^*$ (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.