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International Journal of Mathematics and Mathematical Sciences
Volume 2005, Issue 4, Pages 579-592

Unit groups of cube radical zero commutative completely primary finite rings

Department of Mathematics, University of Transkei, Private Bag X1, Umtata 5117, South Africa

Received 1 July 2004

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.


A completely primary finite ring is a ring R with identity 10 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/JGF(pr) and the characteristic of R is pk, where 1k3, 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=RoUV, 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.