Coprime Numbers

Description

Two numbers (or ring elements) are called coprime if addition and subtraction generates all possible numbers in the Ring.

Definition (General rings)

Two Ideals $I_1,I_2\subseteq R$ of a Ring $R$ are called coprime if $I_1+I_2=R$.

In the case of a Principal Ideal Domain (like $\mathbb{Z}$), this is equivalent with the property that the greatest common divisor is $1$. This apparently follows from the Bezout’s Lemma.

Definition

Let $R$ be a Principal Ideal Domain. Two numbers $a_1,a_2\in R$ are called coprime if $ggT(a_1,a_2)$ is a unit (like $1$).