On structures of the ring of arithmetical functions: prime ideals and beyond
Amartya Goswami, Danielle Kleyn, Kerry Porrill
TL;DR
The paper analyzes the ring $A$ of arithmetical functions under Dirichlet convolution, proving it is neither Noetherian nor Artinian and has infinite Krull dimension. It characterizes the unit structure and local nature with the maximal ideal $m$, and develops a prime-ideals framework built from the principal primes $P_p$ and the families $P_m$ and $J_Q$, establishing how they relate (for example $P_m = \sum_{q|m} P_q$ and $J_Q$ prime, often equal to $P_m$ when the prime complement is finite). It also exhibits infinite ascending chains of primes and a semi-prime non-prime family $P_{m,k}$, together illustrating a rich non-Noetherian landscape. These results enhance understanding of the arithmetic function ring and connect number theory with detailed ideal-theoretic structure.
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.