Chiteng'a John Chikunji

Copyright © 2005 Chiteng'a John Chikunji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.