Some remarks on the ring of arithmetical functions

TL;DR

This paper investigates the structural properties of the ring of arithmetical functions, demonstrating it is neither Noetherian nor Artinian, has infinite Krull dimension, and features various prime ideals.

Contribution

It provides new insights into the algebraic structure of the ring of arithmetical functions, including prime ideal construction and dimension analysis.

Findings

The ring is neither Noetherian nor Artinian.

It has infinite Krull dimension.

The set of associated prime ideals is nonempty.

Abstract

The aim of these notes is to study some of the structural aspects of the ring of arithmetical functions. We prove that this ring is neither Noetherian nor Artinian. Furthermore, we construct various types of prime ideals. We show arithmetic ring has infinite Krull dimension and its associated prime ideal set is nonempty.