On the groups of units of finite commutative chain rings

A finite commutative chain ring is a finite commutative ring whose ideals form a chain. Let R be a finite commutative ring with maximal ideal M and characteristic pn such that R/M≅GF(pr) and pR=Me, e⩽s, where s is the nilpotency of M. When (p−1)∤e, the structure of the group of units R× of R has been determined; it only depends on the parameters p,n,r,e,s. In this paper, we give an algorithmic method which allows us to compute the structure of R× when (p−1)|e; such a structure not only depends on the parameters p,n,r,e,s, but also on the Eisenstein polynomial which defines R as an extension over the Galois ring GR(pn,r). In the case (p−1)∤e, we strengthen the known result by listing a set of linearly independent generators for R×. In the case (p−1)|e but p∤e, we determine the structure of R× explicitly.