Associate

Description

Two elements in a Ring are associated if they differ only by a unit (e.g. $-1$).

Definition

Let $R$ be a Ring and $a,b\in R$. If we have the Divisor-relations $a|b$ and $b|a$ then the elements $a$ and $b$ are called associated or associates zueinander.[^1]

Characterization

If $R$ is an Integral domain, then $a,b\in \mathbb{R}$ are associated i.f.f. there is a unit $\epsilon \in R^{\times }$ with $b=\epsilon a$