On J-Diagrams for the One Groups of Finite Chain Rings

Abstract

Let R be a finite commutative chain ring with invariants p,n,r,k,m. The purpose of this article is to study j-diagrams for the one group H=1+J(R) of R, where J(R)=(π) is Jacobson radical of R. In particular, we prove the existence and uniqueness of j-diagrams for such one group. These j-diagrams help us to solve several problems related to chain rings such as the structure of their unit groups and a group of all symmetries of {πk′}, where k′∣k. The invariants p,n,r,k,m and the Eisenstein polynomial by which R is constructed over its Galois subring determine fully the j-diagram for H.