nLab strictly ordered ring

Idea

A notion of ordered ring for strict orders which are not necessarily linear orders.

A strictly ordered ring is an ring $R$ with a strict order $\lt$ such that

$0 \lt 1$
for all $a \in R$ and $b \in R$ , if $0 \lt a$ and $0 \lt b$ , then $0 \lt a + b$
for all $a \in R$ and $b \in R$ , if $0 \lt a$ and $0 \lt b$ , then $0 \lt a \cdot b$

Every strictly ordered ring is a preordered ring given by the negation of the strict order. In the presence of excluded middle, every strictly ordered ring is a totally preordered ring.

Every linearly ordered ring is a strictly ordered ring.
Every ordered local ring is a strictly ordered ring where every element greater than zero or less than zero is invertible.
Every ordered Kock field is a example of a strictly ordered ring which in the presence of excluded middle is a linearly ordered ring.