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 p n such that R/ M≅GF( p r ) and pR= M e , 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( p n , 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.