NILPOTENT RINGS AND FINITE PRIMARY RINGS WITH CYCLIC GROUPS OF UNITS Cora Stack Abstract. In this paper, we use properties of nilpotent rings to reprove an old theorem of R. Gilmer which classifies finite commutative primary rings having a cyclic group of units. Introduction The following properties of primary rings are required for our proof (see [8]). A finite ring R is primary if its set of zero divisors forms an additive group, or equivalently, if it is an ideal. If we denote the set of zero divisors of the primary ring R by M , then M is the unique maximal ideal of R and hence is the Jacobson radical of R, and is therefore nilpotent. This implies in particular that M i ⊃ M i+1 for each non-zero M i . The quotient field R/M is a finite field called the residue field. Thus R/M is the Galois field GF(p t ) of p t elements, where p is a prime and t a positive integer. The quotient spaces M i /M i+1 may be regarded as vector spaces over the residue field R/M via the action defined by (r + M )(m + M i+1 )= rm + M i+1 , for r ∈ R and m ∈ M i . Moreover, |R| = p tk for some positive integer k. Finally, in case R is commutative, the group of units is the direct product of the p-subgroup 1 + M and a cyclic group of order p t − 1, that is, R ∗ = (1 + M ) × C p t −1 , where C s denotes the cyclic group of order s. Thus the group of units of a finite commutative primary ring is cyclic if and only if the p-subgroup 1+ M is cyclic. 55