nLab pseudo-ordered ring

 Idea

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

Definition

A pseudo-ordered ring or a strictly totally ordered ring is an ring $R$ with a pseudo-order/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 $0 \lt a \cdot b$

 Properties

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.

 Examples

• the integers are a pseudo-ordered ring

• the Dedekind real numbers are a pseudo-ordered ring which in constructive mathematics cannot be proved to be a totally ordered ring.

 See also

• strictly weakly ordered ring
• partially ordered ring
• totally ordered ring
• ordered integral domain
• ordered field