Characterization of the numbers which satisfy the height reducing property

Let $\alpha$ be a complex number. We show that there is a finite subset $F$ of the ring of the rational integers $\mathbb{Z}$, such that $F\left[ \alpha\right] =$\mathbb{Z}$\left[ \alpha\right]$,ifandonlyif$\alpha$ is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. This completes the answer to a question, on the numbers satisfying the height reducing property, posed in [3].