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 be a commutative ring. is an ordered local ring if there is a strict weak order such that
for all and , and implies that ; alternatively, implies that or .
for all and , if and , then
for all , is invertible if and only if or
Every ordered local ring is a local ring with the apartness relation given by
Every ordered local ring has a preorder given by .
Let be the ideal of all non-invertible elements in . Then the quotient ring 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 are an ordered local ring where the nilpotent infinitesimal 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.
Last revised on March 2, 2024 at 15:49:59. See the history of this page for a list of all contributions to it.