  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.

Commutative Algebra;Rings and Algebras