GCD from Generator of Ideal

Theorem
Let $m, n \in \Z$, with either $m \ne 0$ or $n \ne 0$.

Let $I = \left({ m, n }\right)$ be the ideal generated by $m$ and $n$.

Let $d$ be a non-negative generator for the principal ideal $I$.

Then:
 * $\gcd \left\{{m, n}\right\} = d$

where $\gcd \left\{{m, n}\right\}$ denotes the greatest common divisor of $m$ and $n$.

Proof
First we show that such an element $d$ exists.

By Ring of Integers is Principal Ideal Domain there exists a generator $e$ of $I$.

If $e < 0$, then since the units of $\Z$ are $\left\{{ \pm 1 }\right\}$, we have by definition that $-e$ is an associate of $e$.

Therefore by $(3)$ of Principal Ideals in Integral Domain $-e > 0$ is also a generator of $I$.

In particular setting $d = \max\left\{{ e, -e }\right\}$, $d$ is a non-negative generator of $I$.

By Bézout's Lemma, we are required to show that $d$ is the smallest positive integer combination of $m$ and $n$.

By definition:
 * $I = \left\{{ am + bn : a,b \in \Z }\right\}$

Thus we are required to show that $d$ is the smallest positive element of $I$.

Suppose that $d' \le d$ is a positive element of $I$, not larger than $d$.

By hypothesis $d$ generates $I$, so there exists $a \in \Z$ such that $d' = ad$.

Since $d > 0$, we can therefore write $\displaystyle a = \frac {d'} {d} \in \Q$.

Moreover, because $d' > 0$, by $(6)$ of Properties of Ordered Ring we have:
 * $0 = \dfrac 0 d < \dfrac {d'} d$

Using the hypothesis that $d' \le d$, we have the pair of inequalities:
 * $0 < a = \dfrac {d'} d \le 1$

By the definition we have $a \in \Z$, so this shows that $\dfrac {d'} {d} = 1$.

It follows that $d' = d$.

Therefore there is no positive element of $I$ smaller than $d$.