nLab weak local ring

 Idea

A weak notion of local ring in the context of constructive mathematics.

Definition

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.

 Properties

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 weak Heyting field (cf. Richman 2020) or a Johnstone residue field (cf. 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 weak Heyting fields are precisely the weak local rings for which $\approx$ implies equality.

Examples

• Every weak Heyting 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.

Weakly ordered local rings

A weakly ordered local ring is a weak local ring $R$ with a preorder $\leq$ such that

• for all $a \in R$ and $b \in R$, $a \approx b$ if and only if $a \leq b$ and $b \leq a$
• for all $a \in R$, $b \in R$, and $c \in R$, if $a \leq b$, then $a + c \leq b + c$
• for all $a \in R$ and $b \in R$, if $0 \leq a$ and $0 \leq b$, then $0 \leq a \cdot b$

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 Heyting field, and the preorder becomes a partial order.

Every ordered local ring and thus every ordered field is a weakly ordered local ring.

 See also

• local ring

• ordered local ring

References

• Peter Johnstone, Rings, Fields, and Spectra, Journal of Algebra 49 (1977) pp 238-260. doi:10.1016/0021-8693(77)90284-8

• Fred Richman, Laurent series over $\mathbb{R}$. Communications in Algebra, Volume 48, Issue 5, 11 Jan 2020 Pages 1982-1984 [doi:10.1080/00927872.2019.1710166]