## Definition ## An __ordered integral domain__ is a [[totally ordered commutative ring]] which comes with a strict order $\lt$ such that * $0 \lt 1$ * for all elements $a:R$ and $b:R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a + b$ * for all elements $a:R$ and $b:R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$ * for all elements $a:R$ and $b:R$, if $0 \lt \max(a, -a)$ and $0 \lt \max(b, -b)$, then $0 \lt \max(a \cdot b, -a \cdot b)$ ## Examples ## * The [[integers]] are an ordered integral domain. * The [[rational numbers]] are a ordered integral domain * Every [[ordered field]] is a ordered integral domain. ## See also ## * [[ordered field]] * [[unit interval]] * [[Archimedean ordered integral domain]]