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 RR be a commutative ring. RR is an ordered local ring if there is a binary relation <\lt such that

  • for all aRa \in R, ¬(a<a)\neg(a \lt a)

  • for all aRa \in R and bRb \in R, if a<ba \lt b, then ¬(b<a)\neg(b \lt a)

  • for all aRa \in R, bRb \in R, and cRc \in R, if a<ca \lt c, then a<ba \lt b or b<cb \lt c

  • 0<10 \lt 1

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

We define the predicate isPositive\mathrm{isPositive} as

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

With a positivity predicate

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

  • 0 is not positive
  • given element aRa \in R, if aa is positive, then a-a is not positive
  • given elements aRa \in R and bRb \in R; if aa is positive, then either bb is positive or aba - b is positive
  • 1 is positive
  • given elements aRa \in R and bRb \in R, if aa is positive and bb is positive, a+ba + b is positive
  • given elements aRa \in R and bRb \in R, if aa is positive and bb is positive, aba \cdot b is positive
  • for all aRa \in R, aa is invertible if and only if aa is positive or a-a is positive

We define the order relation <\lt as

a<bisPositive(ba)a \lt b \coloneqq \mathrm{isPositive}(b - 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 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 [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.

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.