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 be a commutative ring. is an ordered local ring if there is a binary relation such that
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for all ,
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for all and , if , then
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for all , , and , if , then or
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for all and , if and , then
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for all and , if and , then
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for all , is invertible if and only if or
We define the predicate as
With a positivity predicate
Let be a commutative ring. is an ordered local ring if there is a predicate stating an element is positive, such that
- 0 is not positive
- given element , if is positive, then is not positive
- given elements and ; if is positive, then either is positive or is positive
- 1 is positive
- given elements and , if is positive and is positive, is positive
- given elements and , if is positive and is positive, is positive
- for all , is invertible if and only if is positive or is positive
We define the order relation as
Properties
Every ordered local ring is a local ring with the apartness relation given by
Every ordered local ring has a preorder given by .
Quotient ordered field
Let be the ideal of all non-invertible elements in . Then the quotient ring is an ordered field.
Examples
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Every ordered field is an ordered local ring where every non-positive non-negative element is equal to zero.
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Every ordered Kock field is an ordered local ring.
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The dual numbers are an ordered local ring where the nilpotent infinitesimal 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.