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

Abstract

Let R be a commutative completely primary finite ring with a unique maximal ideal Z(R) such that (Z(R))5 = (0); (Z(R))4 ̸= (0). Then R/Z(R) ∼= GF(p r ) is a finite field of order p r . Let R0 = GR(p kr, pk ) be a Galois ring of order p kr and of characteristic p k for some prime number p and positive integers k, 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 e, f, g and h elements respectively. Then R is of characteristic p k where 1 ≤ k ≤ 5 . In this paper, we investigate and determine the structures of the unit groups of some classes of commutative completely primary finite ring R with pui = p ξ vj = pwk = pyl = 0, where ξ = 2, 3; 1 ≤ i ≤ e, 1 ≤ j ≤ f, 1 ≤ k ≤ g, and 1 ≤ l ≤ h.