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 p(n) such that R/M congruent to GF(p(r)) and pR= M-e, eless than or equal tos, where s is the nilpotency of M. When (p-1)inverted iotae, the structure of the group of units R-x 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-x 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(p(n), r). In the case (p-1)inverted iotae, we strengthen the known result by listing a set of linearly independent generators for R-x. In the case (p-1)\e but p inverted iota e, we determine the structure of R-x explicitly. (C) 2002 Elsevier Science (USA). All rights reserved.