# Arithmetical Rings and Krull Dimension

## TL;DR

This paper characterizes when a commutative arithmetical ring has Krull dimension, linking it to properties of its factor rings such as finite-dimensionality and the absence of certain idempotent ideals.

## Contribution

It provides a necessary and sufficient condition for a commutative arithmetical ring to have Krull dimension based on its factor rings' properties.

## Key findings

- Krull dimension characterized by factor ring properties
- Finite-dimensionality of factor rings is essential
- Absence of idempotent proper essential ideals is necessary

## Abstract

Let $A$ be a commutative arithmetical ring. The ring $A$ has Krull dimension if and only if every factor ring of $A$ is finite-dimensional and does not have idempotent proper essential ideals. The study is supported by Russian Science Foundation (project no. 16-11-10013).