Given an ordered abelian group $A$, let us inductively define the left action $act:\mathbb{N} \times A \to A$ as
$A$ is an Archimedean ordered abelian group if there is a family of dependent terms
$A$ is an Archimedean ordered abelian group if there is a family of dependent terms
The integers are an Archimedean ordered abelian group.
The rational numbers are a Archimedean ordered abelian group.
Every Archimedean ordered integral domain is a Archimedean ordered abelian group.