In this thesis we introduce the ring Int$(\mathbb{Z})$, which consists of polynomials with rational coefficients that take integer values for integers. This ring has different properties from most of the rings studied in commutative algebra. We have focused on the fact that the polynomial with integer values has the property of two generators. The known proofs of this property are rather complicated, since they use strong topological arguments. In this paper we present a constructive proof that uses basic algebraic tools. For a better understanding, we define notions such as the ring, the ideal, Noethererian ring and the Pr ̈ufer domain. For the proof of the two-generator property, we have used the extended Euclidean algorithm, the Skolem property of characterising ideals by their ideals of values, and other necessary assertions and lemmas. Throughout the thesis, we compare the polynomial ring with integer values with the polynomial ring with integer coefficients and illustrate the described properties with examples.
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