Abstract
Let αα be a complex number. We show that there is a finite subset FF of the ring of the rational integers ZZ, such that F[α]=Z[α]F[α]=Z[α], if and only if αα 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).