Unit Groups of a Certain Class of Completely Primary Finite Rings

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(ργ) and the characteristic of R is ργ, where 1 ≤ k ≤ 3, for some prime ρ and positive integer γ. 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 non-gative 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 p2 and p3; and when s = 2, t = 2 and the characteristic of R is p, the structure of the group of units R* of the ring R and its generators have been determined; these depend on the structural matrices (alij) and on the parameters p, k, r, s and t.