Characterisation of the numbers which satisfy the height reducing property

Let α be a complex number. We show that there is a finite subset F of the ring of the rational integers Z, such that F[α]=Z[α], if and only if α 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 Akiyama and Zaïmi (2013).