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

Abstract

What are all rings $R$ for which $R^{\ast }$ (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.

11 pages, 1 figure, to appear in the Journal of Commutative Algebra