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
$0 \neq 1$; 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 $\approx$, defined as $x \approx y$ if and only if $x - y$ is non-invertible. Then Johnstone’s residue fields are precisely the weak local rings for which $\approx$ implies equality.
Every Johnstone residue field is an weak local ring where every non-invertible element is equal to zero.
The dual algebra $\mathbb{R}[\epsilon]/\epsilon^2$ of the MacNeille real numbers $\mathbb{R}$ is a weak local ring where the nilpotent infinitesimal $\epsilon \in \mathbb{R}[\epsilon]/\epsilon^2$ is a non-zero non-invertible element.
For any prime number $p$ and any positive natural number $n$, the prime power local ring $\mathbb{Z}/p^n\mathbb{Z}$ is an weak local ring, whose ideal of non-invertible elements is the ideal $p(\mathbb{Z}/p^n\mathbb{Z})$. The quotient of $\mathbb{Z}/p^n\mathbb{Z}$ by its ideal of non-invertible elements is the finite field $\mathbb{Z}/p\mathbb{Z}$.
Every local ring is a weak local ring with an apartness relation $\#$ such that for all $a \in R$ and $b \in R$, $a \# b$ if and only if $a - b$ is invertible. The negation of $a \# b$ is an equivalence relation which holds if and only if $a - b$ is non-invertible, making every local ring a weak local ring.
A weakly ordered local ring is a weak local ring $R$ with a preorder $\leq$ such that
If additionally, for all $a \in R$ and $b \in R$, $a \approx b$ implies that $a = b$, 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.