nLab ordered local ring

Redirected from "ordered local rings".

Idea

In the same way that a local ring is a Heyting field whose apartness relation is not tight, an ordered local ring is an ordered field whose strict weak order is not necessarily connected.

Let $R$ be a commutative ring. $R$ is an ordered local ring if there is a strict weak 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 $0 \lt a \cdot b$
for all $a \in R$ , $a$ is invertible if and only if $a \lt 0$ or $0 \lt a$

Every ordered local ring is a local ring with the apartness relation given by

a \# b \coloneqq a \lt b \vee b \lt a

Every ordered local ring has a preorder given by $a \leq b \coloneqq \neg (b \lt a)$ .

Let $D$ be the ideal of all non-invertible elements in $R$ . Then the quotient ring $R/D$ is an ordered field.

Every ordered discrete field is an ordered local ring where the order relation satisfies trichotomy.
Every ordered Heyting field is an ordered local ring where every non-positive non-negative element is equal to zero.
Every ordered Kock field is an ordered local ring in which every non-zero element is positive or negative.
The dual numbers $\mathbb{R}[\epsilon]/\epsilon^2$ are an ordered local ring where the nilpotent infinitesimal $\epsilon \in \mathbb{R}[\epsilon]/\epsilon^2$ is a non-zero non-positive non-negative element.

Ordered local rings are important for modeling notions of infinitesimals and infinite elements, including both the non-invertible infinitesimals common in synthetic differential geometry and the invertible infinitesimals whose reciprocals are the infinite elements common in nonstandard analysis.