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