  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.
