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

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 $\text{ms}A$ be the algebra consisting of all complex periodic functions on $\mathbb{Z}$ with the uniform norm. Then the polyadic topological ring can be defined as the ring of all characters $\text{ms}A\to \text{C}$ with convolution operations and the Gelfand topology.