An Elementary Proof of the Two-Generator Property for the Ring of Integer-Valued Polynomials

The ring Int(Z) of integer-valued polynomials has the two-generator property, which means that every finitely generated ideal may be generated by two elements. As the known proofs of this fact are rather complicated using strong topological arguments, we propose here a constructive proof obtained by means of elementary tools. Along the way, we also obtain constructive proofs of two other well-known facts: the finitely generated ideals of Int(Z) are characterized by their ideals of values and Int(Z) is a two-dimensional Prufer domain.