symmetric monoidal (∞,1)-category of spectra
A weak notion of local ring in the context of constructive mathematics.
A weak local ring is a commutative ring such that
; and
the sum of two non-invertible elements is non-invertible
These are the same as local rings in classical mathematics, but are a weaker (and thus more general) notion than local rings in constructive mathematics.
The non-invertible elements in a weak local ring form an ideal. Thus, the quotient of a weak local ring by its ideal of non-invertible elements form a residue field in the sense of Johnstone 1977.
Every weak local ring has an equivalence relation , defined as if and only if is non-invertible. Then Johnstone’s residue fields are precisely the weak local rings for which implies equality.
Every Johnstone residue field is an weak local ring where every non-invertible element is equal to zero.
The dual algebra of the MacNeille real numbers is a weak local ring where the nilpotent infinitesimal is a non-zero non-invertible element.
For any prime number and any positive natural number , the prime power local ring is an weak local ring, whose ideal of non-invertible elements is the ideal . The quotient of by its ideal of non-invertible elements is the finite field .
Every local ring is a weak local ring with an apartness relation such that for all and , if and only if is invertible. The negation of is an equivalence relation which holds if and only if is non-invertible, making every local ring a weak local ring.
A weakly ordered local ring is a weak local ring with a preorder such that
If additionally, for all and , implies that , then a weakly ordered local ring becomes a weakly ordered residue field, and the preorder becomes a partial order.
Every ordered local ring and thus every ordered field is a weakly ordered local ring.
Last revised on December 8, 2022 at 23:11:30. See the history of this page for a list of all contributions to it.