Since every ordered local integral domain$R$ has characteristic zero, the positive integers $\mathbb{Z}_+$ are a subset of $R$, with injection$i:\mathbb{Z}_+ \hookrightarrow R$. An Archimedean ordered local integral domain is an ordered local integral domain which satisfies the archimedean property: for all elements $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then there exists a positive integer $n \in \mathbb{Z}_+$ such that such that $a \lt i(n) \cdot b$.