In 1963, Gilmer characterized
all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and
Fischer generalized these results to rings with minimum condition. In the present
paper these results are extended to semiperfect rings and generalizations of the three
theorems are obtained. It is shown that a semiperfect ring with cyclic group of units
is finite and is either commutative or is the direct sum of a commutative ring and the
2 ×2 upper triangular matrix ring over the field of two elements. Let R be
semiperfect with an abelian group of units. It is shown that R is finite if either the
group of units is finite or the group of units is finitely generated and the Jacobson
radical is nil.
The proofs of all these results depend on our main theorem: The structure of a
semiperfect ring R with an abelian group of units is described completely up to the
structure of commutative local rings. (That is commutative rings with a unique
maximal ideal.) The groups of units of these local rings are shown to be direct factors
of the group of units of R.
