nLab pseudo-ordered ring



(0,1)-Category theory



A notion of ordered ring for pseudo-orders/strict total orders.


A pseudo-ordered ring or a strictly totally ordered ring is an ring RR with a pseudo-order/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


Every pseudo-ordered ring is a partially ordered ring given by the negation of the strict total order. In the presence of excluded middle, every pseudo-ordered ring is a totally ordered ring.

Pseudo-ordered rings may have zero divisors. The pseudo-ordered rings which do not have zero divisors are ordered integral domains.


 See also

Last revised on February 1, 2024 at 19:00:05. See the history of this page for a list of all contributions to it.