Unit groups of cube radical zero commutative 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(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 (ai j) and on the parameters p, k, r, and s.