Fermat's Little Theorem and Euler's Theorem in a class of rings

TL;DR

This paper extends Fermat's Little Theorem and Euler's Theorem from the ring of integers modulo n to a broader class of rings satisfying certain mild conditions, broadening their applicability.

Contribution

It introduces a generalization of classical number theory theorems to new algebraic structures beyond traditional rings of integers.

Findings

Theorems are valid in a wider class of rings.

Conditions for the extension are explicitly characterized.

Potential applications in algebra and number theory are discussed.

Abstract

Considering $\mathbb{Z}_n$ the ring of integers modulo $n$, the classical Fermat-Euler theorem establishes the existence of a specific natural number $\varphi(n)$ satisfying the following property: $ x^{\varphi(n)}=1%\hspace{1.0cm}\text{for all}\hspace{0.2cm}x\in \mathbb{Z}_n^*, $ for all $x$ belonging to the group of units of $\mathbb{Z}_n$. In this manuscript, this result is extended to a class of rings that satisfies some mild conditions.