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 weak order is not necessarily connected.

Definition

Let RR be a commutative ring. RR is an ordered local ring if there is a strict weak order <\lt such that

  • 0<10 \lt 1

  • for all aRa \in R and bRb \in R, 0<a0 \lt a and 0<b0 \lt b implies that 0<a+b0 \lt a + b; alternatively, 0<a+b0 \lt a + b implies that 0<a0 \lt a or 0<b0 \lt b.

  • for all aRa \in R and bRb \in R, if 0<a0 \lt a and 0<b0 \lt b, then 0<ab0 \lt a \cdot b

  • for all aRa \in R, aa is invertible if and only if a<0a \lt 0 or 0<a0 \lt a

Properties

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

a#ba<bb<aa \# b \coloneqq a \lt b \vee b \lt a

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

Quotient ordered field

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

 Examples

  • 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 [ϵ]/ϵ 2\mathbb{R}[\epsilon]/\epsilon^2 are an ordered local ring where the nilpotent infinitesimal ϵ[ϵ]/ϵ 2\epsilon \in \mathbb{R}[\epsilon]/\epsilon^2 is a non-zero non-positive non-negative element.

 In analysis

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.

See also

Last revised on March 2, 2024 at 15:49:59. See the history of this page for a list of all contributions to it.