# characterizations of integral

###### Theorem.

Let $R$ be a subring of a field $K$,  $1\in R$  and let $\alpha$ be a non-zero element of $K$.  The following conditions are equivalent:

1. 1.

$\alpha$ is integral over $R$.

2. 2.

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

3. 3.

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

4. 4.

$\alpha^{-1}R[\alpha^{-1}]=R[\alpha^{-1}]$.

Proof.  Supposing the first condition 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 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 $R$.

## References

• 1 Emil Artin: .  Lecture notes.  Mathematisches Institut, Göttingen (1959).
Title characterizations of integral CharacterizationsOfIntegral 2013-03-22 14:56:54 2013-03-22 14:56:54 pahio (2872) pahio (2872) 11 pahio (2872) Theorem msc 12E99 msc 13B21 characterisations of integral