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(ρ^γ) and the characteristic of R is ρ^γ, where 1 ≤ k ≤ 3, for some prime ρ and positive integer γ. Let R_o = GR(p^kr, p^k) be a Galois subring of R and let the annihilator of J be J² so that R = R_o ⊕U ⊕V , where U and V are finitely generated R_o-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 (a^l_ij) and on the parameters p, k, r, s and t.