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

Let R0=GRpkr,pk be a Galois maximal subring of  R  so that R=R0⊕U⊕V⊕W⊕Y, where U,V,W, and Y are R0/pR0 spaces considered as R0-modules, generated by the sets u1,⋯,ue,v1,⋯,vf,w1,⋯,wg, and y1,⋯,yh, respectively. Then,  R  is a completely primary finite ring with a Jacobson radical ZR such that ZR5=0 and  ZR4≠0. The residue field R/ZR is a finite field GFpr for some prime p and positive integer  r. The characteristic of R is  pk, 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  pψui=0, ψ=2,3,4; pζvj=0, ζ=2,3; pwk=0, and  pyl=0; 1≤i≤e, 1≤j≤f, 1≤k≤g, and  1≤l≤h.