Classification of chain rings

Abstract

An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants p,n,r,k,k′,m up to isomorphism by finite commutative chain rings (k′=1). Thus the classification of chain rings is reduced to that of finite commutative chain rings.

Keywords

• local ring
• chain ring
• galois ring
• p-adic field
• isomorphism class