nLab weak local ring

Contents

 Idea

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

Definition

A weak local ring is a commutative ring such that

  • 010 \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 xyx \approx y if and only if xyx - 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 [ϵ]/ϵ 2\mathbb{R}[\epsilon]/\epsilon^2 of the MacNeille real numbers \mathbb{R} is a weak local ring where the nilpotent infinitesimal ϵ[ϵ]/ϵ 2\epsilon \in \mathbb{R}[\epsilon]/\epsilon^2 is a non-zero non-invertible element.

  • For any prime number pp and any positive natural number nn, the prime power local ring /p n\mathbb{Z}/p^n\mathbb{Z} is an weak local ring, whose ideal of non-invertible elements is the ideal p(/p n)p(\mathbb{Z}/p^n\mathbb{Z}). The quotient of /p n\mathbb{Z}/p^n\mathbb{Z} by its ideal of non-invertible elements is the finite field /p\mathbb{Z}/p\mathbb{Z}.

  • Every local ring is a weak local ring with an apartness relation #\# such that for all aRa \in R and bRb \in R, a#ba \# b if and only if aba - b is invertible. The negation of a#ba \# b is an equivalence relation which holds if and only if aba - 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 RR with a preorder \leq such that

  • for all aRa \in R and bRb \in R, aba \approx b if and only if aba \leq b and bab \leq a
  • for all aRa \in R, bRb \in R, and cRc \in R, if aba \leq b, then a+cb+ca + c \leq b + c
  • for all aRa \in R and bRb \in R, if 0a0 \leq a and 0b0 \leq b, then 0ab0 \leq a \cdot b

If additionally, for all aRa \in R and bRb \in R, aba \approx b implies that a=ba = 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

References

Last revised on August 19, 2024 at 14:51:24. See the history of this page for a list of all contributions to it.