Greatest Common Divisor for Integers

This definition of the greatest common denominator depends on the fact that $\mathbb{Z}$ is totally ordered. A modification is made to define the $\gcd$ for general rings, however this generalisation is not entirely consistent with the above definition, which is why this is a different note.

This inconsistency is characterised in the following theorem.

Proof

By the definition of divisibility, because $n\times 0=0$, $n\mid 0$ for any integer $n$. Hence there is no greatest such integer.

By the general ring definition, it is clear that $0\mid 0$, and if $n\mid 0$, that is we have some other common divisor, then $n\mid 0$ so $0$ is the greatest such divisor.

Suppose either $a$ or $b$ are non-zero and let $gcd(a,b)=d$ by the integer definition. Then because all GCDs are associates in an integral domain, we have in the ring definition that $gcd(a,b)=\{-d,d\}$ because the only units of the integers are $1$ and $-1$.