nLab ordered integral domain

Ordered integral domains

Ordered integral domains


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’.


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

  • 0<10 \lt 1

  • for all aRa \in R and bRb \in R, 0<a0 \lt a and 0<b0 \lt b implies that 0<a+b0 \lt a + b; alternatively, 0<a+b0 \lt a + b implies that 0<a0 \lt a or 0<b0 \lt b.

  • for all aRa \in R and bRb \in R, if 0<a0 \lt a and 0<b0 \lt b, then 0<ab0 \lt a \cdot b


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.


Every ordered integral domain must have characteristic 00, since we can prove by induction that n>0n \gt 0 for every positive natural number nn.

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.


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

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

Last revised on February 21, 2024 at 06:55:30. See the history of this page for a list of all contributions to it.