nLab ordered integral domain

Idea

An ordered integral domain is an integral domain equipped with a compatible strict total order.

Note that while the adjective ‘ordered’ usually refers to a total order, which by default is non-strict, it is traditionally used more strictly when placed before ‘integral domain’.

Definition

An ordered integral domain is an integral domain $R$ equipped with a strict total order $\lt$ such that:

• $0 \lt 1$

• for all $a \in R$ and $b \in R$, $0 \lt a$ and $0 \lt b$ implies that $0 \lt a + b$; alternatively, $0 \lt a + b$ implies that $0 \lt a$ or $0 \lt b$.

• for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$

Examples

The integral domain $\mathbb{R}$ of real numbers, the integral domain $\mathbb{Z}$ of integers, and the integral domain $\mathbb{Q}$ of rational numbers are all ordered integral domains.

Properties

Every ordered integral domain must have characteristic $0$, since we can prove by induction that $n \gt 0$ for every positive natural number $n$.

The archimedean ordered integral domain are precisely the integral subdomains of the integral domain of real numbers.

Every localization of the integral domain of integers away from a subset not containing zero is a dense linear order, and the Dedekind completion of the resulting integral domain is the integral domain of real numbers. In particular, the Dedekind completions of the dyadic rational numbers and the decimal rational numbers are both the real numbers.

Related concepts

• pseudo-ordered ring

• ordered field

• protoring

References

The first characterization of the integers as an ordered integral domain appeared in:

• Hermann Grassmann, Lehrbuch der Arithmetik für höhere Lehranstalten, Berlin: Enslin, 1861. (Google Books)

though the name “ordered integral domain” does not appear in the text.