The structure of finite local principal ideal rings

TL;DR

This paper characterizes the structure of finite local principal ideal rings, showing they are determined by their maximal ideal's properties, specifically its principality and nilpotency index.

Contribution

It provides a complete description of finite local PIRs, extending understanding of their structure based on maximal ideal characteristics.

Findings

Finite local PIRs have a structure determined by their maximal ideal.

The maximal ideal in such rings is principal with finite nilpotency index.

The paper characterizes all finite local PIRs explicitly.

Abstract

A ring $R$ is called a PIR, if each ideal of $R$ is a principal ideal. An local ring $(R,\mf{m)}$ is a artinian PIR if and only if its maximal ideal $\mf{m}$ is principal and has finite nilpotency index. In this paper, we determine the structure of a finite local PIR.