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

Abstract

Let R0 = GR(pkr, 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 Z(R) such that (Z(R))5 = (0) and (Z(R))4 ≠ (0). The residue field R/Z(R) is a finite field GF(pr) 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.