Let S̲\documentclass[12pt]{minimal}
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\begin{document}$$\underline{S}$$\end{document} be an arbitrary subset of Rn\documentclass[12pt]{minimal}
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\begin{document}$$R^n$$\end{document} where R is a domain with the field of fractions K\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {K}$$\end{document}. Denote the ring of polynomials in n variables over K\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {K}$$\end{document} by K[x̲]\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {K}[\underline{x}] $$\end{document}. The ring of integer-valued polynomials over S̲\documentclass[12pt]{minimal}
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\begin{document}$$\underline{S}$$\end{document}, denoted by Int(S̲,R)\documentclass[12pt]{minimal}
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\begin{document}$$(\underline{S},R)$$\end{document}, is defined as the set of the polynomials of K[x̲]\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {K}[\underline{x}] $$\end{document}, which maps S̲\documentclass[12pt]{minimal}
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\begin{document}$$\underline{S}$$\end{document} to R. In this article, we study the irreducibility of the polynomials of Int(S̲,R)\documentclass[12pt]{minimal}
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\begin{document}$$(\underline{S},R)$$\end{document} for the first time in the case when R is a Unique Factorization Domain. We also show that our results remain valid when R is a Dedekind domain or, sometimes, any domain.
