On the Carmichael rings, Carmichael ideals and Carmichael polynomials

TL;DR

This paper characterizes Carmichael rings and ideals, generalizes Korselt's criterion, and studies Carmichael polynomials over finite fields, revealing their infinite yet sparse nature.

Contribution

It provides a structural classification of Carmichael rings and extends Korselt's criterion to Dedekind domains, also analyzing Carmichael polynomials over finite fields.

Findings

Carmichael rings are structurally characterized.

A generalized Korselt's criterion for Carmichael ideals is established.

Infinitely many Carmichael polynomials exist with zero density.

Abstract

Motivated by Carmichael numbers, we say that a finite ring $R$ is a Carmichael ring if $a^{|R|}=a$ for any $a \in R$. We then call an ideal $I$ of a ring $R$ as a Carmichael ideal if $R/I$ is a Carmichael ring, and a Carmichael element of $R$ means it generates a Carmichael ideal. In this paper, we determine the structure of Carmichael rings and prove a generalization of Korselt's criterion for Carmichael ideals in Dedekind domains. We also study Carmichael elements of polynomial rings over finite fields (called Carmichael polynomials) by generalizing various classical results. For example, we show that there are infinitely many Carmichael polynomials but they have zero density.