On structures of the ring of arithmetical functions: prime ideals and beyond

TL;DR

This paper investigates the structural properties of the ring of arithmetical functions, demonstrating its non-Noetherian, non-Artinian nature, infinite Krull dimension, and constructing various prime and semi-prime ideals.

Contribution

It provides new insights into the prime ideal structure and dimensionality of the ring of arithmetical functions, including examples of semi-prime but not prime ideals.

Findings

The ring is neither Noetherian nor Artinian.

It has infinite Krull dimension.

Constructed examples of prime and semi-prime ideals.

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 also give an example of a semi-prime ideal that is not prime. We show that the ring of arithmetical functions has infinite Krull dimension.