Classification of Unit Groups of Five Radical Zero Completely Primary Finite Rings Whose First and Second Galois Ring Module Generators Are of the Order p^k, k = 2, 3, 4

Abstract

Let R 0 = G R p k r , p k be a Galois maximal subring of R so that R = R 0 ⊕ U ⊕ V ⊕ W ⊕ Y , where U , V , W , and Y are R 0 / p R 0 spaces considered as R 0 -modules, generated by the sets u 1 , ⋯ , u e , v 1 , ⋯ , v f , w 1 , ⋯ , w g , and y 1 , ⋯ , y h , respectively. Then, R is a completely primary finite ring with a Jacobson radical Z R such that Z R 5 = 0 and Z R 4 ≠ 0 . The residue field R / Z R is a finite field G F p r for some prime p and positive integer r . The characteristic of R is p k , where k is an integer such that 1 ≤ k ≤ 5 . In this paper, we study the structures of the unit groups of a commutative completely primary finite ring R with p ψ u i = 0 , ψ = 2 , 3 , 4 ; p ζ v j = 0 , ζ = 2 , 3 ; p w k = 0 , and p y l = 0 ; 1 ≤ i ≤ e , 1 ≤ j ≤ f , 1 ≤ k ≤ g , and 1 ≤ l ≤ h .