Sobre los Enteros de Eisenstein módulo n y sus elementos primos

Abstract

This work studies the Eisenstein integer ring Z[w], as a Euclidean domain, principal ideal domain, and unique factorization domain. A complete classification of its primes is provided according to the behavior of the rational prime p mod 3. Building on this foundation, the quotient ring E_n ≅ Z[w]/nZ[w] of Eisenstein integers modulo n is constructed, which by the Chinese Remainder Theorem decomposes as a product of Eisenstein rings modulo prime powers. The ring E_{p^k} is studied, characterizing when it is a local ring, computing the order of its unit group in terms of p mod 3, and characterizing the values of n for which E_n is a Von Neumann local ring (VNL), following the results of Alkam and Abuosba.