nLab linearly ordered ring

 Idea

A notion of ordered ring for linear orders.

Definition

A linearly ordered ring is an ring $R$ with a linear 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 $0 \lt a \cdot b$

 Properties

Every linearly ordered ring is a partially ordered ring given by the negation of the linear order. In the presence of excluded middle, every linearly ordered ring is a totally ordered ring.

Linearly ordered rings may have zero divisors. The linearly ordered rings which do not have zero divisors are ordered integral domains.

 Examples

• the integers are a linearly ordered ring

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

 See also

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