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 $A$.

The following statements are equivalent:

$(1): \quad a$ is a unit of $A$

$(2): \quad a$ is a unit of $B$

Proof

1 implies 2

Follows from Ring Homomorphism Preserves Units.

$\Box$

2 implies 1

Let $a$ be a unit of $B$.

Let $P \in A \sqbrk x$ be a monic polynomial with $\map P {1 / a} = 0$.

Let $n$ be its degree and $\map P x = x^n + \map Q x$.

Then $1 + a^n \map Q {1 / a} = 0$.

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

Thus $a$ is a unit of $A$, with inverse $-a^{n - 1} \map Q {1 / a}$.

$\blacksquare$