characterizations of integral
Theorem.
Let $R$ be a subring of a field $K$, $1\in R$ and let $\alpha $ be a nonzero element of $K$. The following conditions are equivalent^{}:

1.
$\alpha $ is integral over $R$.

2.
$\alpha $ belongs to $R[{\alpha}^{1}]$.

3.
$\alpha $ is unit of $R[{\alpha}^{1}]$.

4.
${\alpha}^{1}R[{\alpha}^{1}]=R[{\alpha}^{1}]$.
Proof. Supposing the first condition that an equation
$${\alpha}^{n}+{a}_{1}{\alpha}^{n1}+\mathrm{\dots}+{a}_{n1}\alpha +{a}_{n}=0,$$ 
with ${a}_{j}$’s belonging to $R$, holds. Dividing both by ${\alpha}^{n1}$ gives
$$\alpha ={a}_{1}{a}_{2}{\alpha}^{1}\mathrm{\dots}{a}_{n}{\alpha}^{n+1}.$$ 
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 $R$.
References
 1 Emil Artin: . Lecture notes. Mathematisches Institut, Göttingen (1959).
Title  characterizations of integral 

Canonical name  CharacterizationsOfIntegral 
Date of creation  20130322 14:56:54 
Last modified on  20130322 14:56:54 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  11 
Author  pahio (2872) 
Entry type  Theorem 
Classification  msc 12E99 
Classification  msc 13B21 
Synonym  characterisations of integral 