On the factor rings of Z[n3]

We give a description of the structure of factor rings of $Z\left[\sqrt[3]{n}\right]$ where (without loss of generality) n is a positive integer which is not a cube. For example, we prove that $Z\left[\sqrt[3]{n}\right]/(a+b\sqrt[3]{n}+c\sqrt[3]{n^2})$ is isomorphic to the ring of integers modulo $\left|N\right|$, if $\gcd (b^2-ac,N)=1$ where $N=a^3+n{b}^3+n^2{c}^3-3nabc$ is the norm of the generator. We also characterize the structure of these factor rings for others integers $a,b,c$. Finally, we describe $Z\left[\sqrt[3]{n}\right]/I$ for certain non-principal ideals I. We also present many corollaries regarding irreducible and prime elements in $Z\left[\sqrt[3]{n}\right]$ and give numerous examples.