ON THE CARMICHAEL RINGS, CARMICHAEL IDEALS AND CARMICHAEL POLYNOMIALS

Motivated by Carmichael numbers, we say that a finite ring R is a Carmichael ring if a(vertical bar R vertical bar) = a for any a is an element of R. We then call an ideal I of a ring R 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 extend several results from the number field case to the function field case. In particular, we study Carmichael elements of polynomial rings over finite fields (called Carmichael polynomials) by generalizing some classical results. For example, we show that there are infinitely many Carmichael polynomials but they have zero density.