Suppose that $R$ is a partially ordered commutative ring. The order behaves nicely: We have $0 \leq 1$. If $a \leq b$ and $c > 0$, then $a + c \leq b + c$ and $a \cdot c \leq b \cdot c$. For $n \in \mathbb{Z}$ define $\hat{n} = 1 + \cdots + 1$(n terms), where $1$ is the ring's multiplicative identity. The property of the ring is:
For all $r \in R$ such that $r > 0$ is an $n \in \mathbb{Z}$ such that either $1 < \hat{n} \cdot r$ or $\hat{n} \cdot r \nleq 1$.
The first case is like the Archimedean property. The second case says there are none of the usual infinitesimals.
If this property has a commonly used name, I would like to know it. Any references would be appreciated.