Ring Element is Unit iff Unit in Integral Extension

Theorem
let $A$ be a commutative ring with unity.

Let $a \in A$.

Let $B$ be an integral ring extension of $A$.


 * 1) $a$ is invertible in $A$
 * 2) $a$ is invertible in $B$
 * 1) $a$ is invertible in $B$

1 implies 2
Follows from Ring Homomorphism Preserves Invertible Elements.

2 implies 1
Let $a$ be invertible in $B$.

Let $P \in A[x]$ be a monic polynomial with $P(1/a) = 0$.

Let $n$ be its degree and $P(x) = x^n + Q(x)$.

Then $1 + a^n Q(1/a) = 0$.

Note that $a^{n-1} Q(1/a) \in A$.

Thus $a$ is invertible in $A$, with inverse $-a^{n-1} Q(1/a)$.