Proof. Supposing the first condition means that an equation $$\alpha^n+a_1\alpha^{n-1}+\ldots+a_{n-1}\alpha+a_n = 0,$$ with $a_j$ s belonging to $R$ holds. Dividing both sides by $\alpha^{n-1}$ gives $$\alpha = -a_1-a_2\alpha^{-1}-\ldots-a_n\alpha^{-n+1}.$$ One sees that $\alpha$ belongs to the ring $R[\alpha^{-1}]$ even being a unit of this (of course $\alpha^{-1}\in R[\alpha^{-1}]$ . Therefore also the principal ideal $\alpha^{-1}R[\alpha^{-1}]$ of the ring $R[\alpha^{-1}]$ coincides with this ring. Conversely, the last circumstance implies that $\alpha$ is integral over $R$

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Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).