nLab
ordered local ring
Contents
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 order is not necessarily linear .

Definition
With an order relation
Let $R$ be a commutative ring . $R$ is an ordered local ring if there is a binary relation $\lt$ such that

for all $a \in R$ , $\neg(a \lt a)$

for all $a \in R$ and $b \in R$ , if $a \lt b$ , then $\neg(b \lt a)$

for all $a \in R$ , $b \in R$ , and $c \in R$ , if $a \lt c$ , then $a \lt b$ or $b \lt c$

$0 \lt 1$

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 $0 \lt a \cdot b$

for all $a \in R$ , $a$ is invertible if and only if $a \lt 0$ or $0 \lt a$

We define the predicate $\mathrm{isPositive}$ as

$\mathrm{isPositive}(a) \coloneqq 0 \lt a$

With a positivity predicate
Let $R$ be a commutative ring . $R$ is an ordered local ring if there is a predicate $\mathrm{isPositive}$ stating an element $a$ is positive, such that

0 is not positive
given element $a \in R$ , if $a$ is positive, then $-a$ is not positive
given elements $a \in R$ and $b \in R$ ; if $a$ is positive, then either $b$ is positive or $a - b$ is positive
1 is positive
given elements $a \in R$ and $b \in R$ , if $a$ is positive and $b$ is positive, $a + b$ is positive
given elements $a \in R$ and $b \in R$ , if $a$ is positive and $b$ is positive, $a \cdot b$ is positive
for all $a \in R$ , $a$ is invertible if and only if $a$ is positive or $-a$ is positive
We define the order relation $\lt$ as

$a \lt b \coloneqq \mathrm{isPositive}(b - a)$

Properties
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 $a \leq b \coloneqq \neg (b \lt a)$ .

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

Examples
Every ordered 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.

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.

See also
Last revised on January 12, 2023 at 17:04:49.
See the history of this page for a list of all contributions to it.