Recapturing the Structure of Group of Units of Any Finite Commutative Chain Ring

A finite ring with an identity whose lattice of ideals forms a unique chain is called ring. Let R be commutative invariants p,n,r,k,m. It known that Eisenstein extension degree k Galois S=GR(pn,r). If p−1 does not divide k, the structure unit group U(R) known. The case (p−1)∣k was partially considered by M. Luis (1991) providing counterexamples demonstrated results Ayoub failed to capture direct decomposition U(R). In this article, we manage determine when fixing Ayoub’s approach. We also sharpen our introducing system generators for and enumerating same order.