Pr\"ufer's Ideal Numbers as Gelfand's maximal Ideals

TL;DR

This paper explores the deep connections between polyadic arithmetics and the theory of commutative Banach algebras, focusing on the structure of characters and maximal ideals.

Contribution

It establishes a link between polyadic topological rings and Gelfand's maximal ideals within the framework of complex periodic functions.

Findings

Polyadic topological ring is characterized as the set of all characters of the algebra.

The algebra of complex periodic functions on integers is used to model polyadic arithmetics.

Connections between polyadic structures and Gelfand theory are demonstrated.

Abstract

Polyadic arithmetics is a branch of mathematics related to $p$--adic theory. The aim of the present paper is to show that there are very close relations between polyadic arithmetics and the classic theory of commutative Banach algebras. Namely, let $\ms A$ be the algebra consisting of all complex periodic functions on $\Z$ with the uniform norm. Then the polyadic topological ring can be defined as the ring of all characters $\ms A\to\C$ with convolution operations and the Gelfand topology.