On the groups of units of finite commutative chain rings

Abstract

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, ≤s, where s is the nilpotency of M. When (P - 1) ł 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) ł e, we strengthen the known result by listing a set of linearly independent generators for R×. In the case (p - 1) e but p ł e, we determine the structure of R× explicitly. © 2002 Elsevier Science (USA). All rights reserved.