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 J 3 = ( 0 ) and J 2 ≠( 0 ) . Then R / J ≅ G F ( p r ) and the characteristic of R is p k , where 1 ≤ k ≤ 3 , for some prime p and positive integer r . Let R o = G R ( p k r , p k ) be a Galois subring of R and let the annihilator of J be J 2 so that R = R o ⊕ U ⊕ V , where U and V are finitely generated R o -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 ( a i j ) and on the parameters p , k , r , and s .