International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 4, Pages 579-592
doi:10.1155/IJMMS.2005.579

Unit groups of cube radical zero commutative completely primary finite rings

Abstract

A completely primary finite ring is a ring R with identity 1≠0 whose subset of all its zero-divisors forms the unique maximal ideal J. Let R be a commutative completely primary finite ring with the unique maximal ideal J such that J3=(0) and J2≠(0). Then R/J≅GF(pr) and the characteristic of R is pk, where 1≤k≤3, for some prime p and positive integer r. Let Ro=GR(pkr,pk) be a Galois subring of R and let the annihilator of J be J2 so that R=Ro⊕U⊕V, where U and V are finitely generated Ro-modules. Let nonnegative integers s and t be numbers of elements in the generating sets for U and V, respectively. When s=2, t=1, and the characteristic of R is p; and when t=s(s+1)/2, for any fixed s, the structure of the group of units R∗ of the ring R and its generators are determined; these depend on the structural matrices (aij) and on the parameters p, k, r, and s.