Classication of Units of Five Radical Zero Completely Primary Finite Rings with Variant Orders of Second Galois Ring Module Generators

LetRbe a commutative completely primary nite ring with a unique maximal idealZ(R) suchthat (Z(R))5= (0); (Z(R))4̸= (0). ThenR=Z(R)∼=GF(pr) is a nite eld of orderpr:LetR0=GR(pkr; pk) be a Galois ring of orderpkrand of characteristicpkfor some prime numberpand positive integersk; rso thatR=R0⊕U⊕V⊕W⊕Y, whereU; V; WandYareR0=pR0- spaces considered asR0modules generated bye; f; gandhelements respectively.ThenRis of characteristicpkwhere 1≤k≤5 . In this paper, we investigate and determine thestructures of the unit groups of some classes of commutative completely primary nite ringRwithpui=p�vj=pwk=pyl= 0;where�= 2;3; 1≤i≤e;1≤j≤f;1≤k≤g;and 1≤l≤h.