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. $\alpha$ 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

$${\alpha }^n+a_1{\alpha }^{n-1}+\mathrm{\dots }+a_{n-1}\alpha +a_n=0,$$

with $a_j$’s belonging to $R$, holds.  Dividing both by ${\alpha }^{n-1}$ gives

$$\alpha =-a_1-a_2{\alpha }^{-1}-\mathrm{\dots }-a_n{\alpha }^{-n+1}.$$

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 $R$.