feat(ring_theory): Fractional ideals #2062
Merged
Conversation
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
semorrison
reviewed
Feb 26, 2020
jcommelin
reviewed
Feb 26, 2020
| variables (R : Type u) [integral_domain R] (S : set R) [is_submonoid S] | ||
|
|
||
| /-- A submodule `I` is a fractional ideal if `a I ⊆ R` for some `a ≠ 0`. -/ | ||
| def is_fractional (I : submodule R (localization R S)) := |
There was a problem hiding this comment.
When it comes to applying this theory, this definition would benefit enormously from using a characteristic predicate for localisations. For example, it will now be hard (i.e. impossible) to talk about fractional ideals in rat wrt to int. Because rat is not of the form localization int _.
@101damnations is working on bringing this characteristic predicate to mathlib. I hope that these two PR will play well together.
There was a problem hiding this comment.
I expect that changing localization ℤ (non_zero_divisors ℤ) to, let's say a typeclass has_localization ℤ ℚ, shouldn't have too much impact on this PR, since most of it is only concerned with is_integer and its lemmas.
Co-Authored-By: Scott Morrison <scott@tqft.net>
(but both reduce to reflexivity anyway)
Since that is how the definition of `I / J` is traditionally done, but it is not as convenient to work with, I didn't change the definition but added a lemma stating the two are equivalent
(it got broken because I merged changes incorrectly)
jcommelin
added
the
ready-to-merge
anrddh
pushed a commit
to anrddh/mathlib
that referenced
this pull request
May 15, 2020
* Some WIP work on fractional ideals * Fill in the `sorry` * A lot of instances for fractional_ideal * Show that an invertible fractional ideal `I` has inverse `1 : I` * Cleanup and documentation * Move `has_div submodule` to algebra_operations * More cleanup and documentation * Explain the `non_zero_divisors R` in the `quotient` section * whitespace Co-Authored-By: Scott Morrison <scott@tqft.net> * `has_inv` instance for `fractional_ideal` * `set.univ.image` -> `set.range` * Fix: `mem_div_iff.mpr` should be `mem_div_iff.mp` (but both reduce to reflexivity anyway) * Add `mem_div_iff_smul_subset` Since that is how the definition of `I / J` is traditionally done, but it is not as convenient to work with, I didn't change the definition but added a lemma stating the two are equivalent * whitespace again (it got broken because I merged changes incorrectly) * Fix unused argument to `inv_nonzero` Co-authored-by: Scott Morrison <scott@tqft.net> Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>
anrddh
pushed a commit
to anrddh/mathlib
that referenced
this pull request
May 16, 2020
* Some WIP work on fractional ideals * Fill in the `sorry` * A lot of instances for fractional_ideal * Show that an invertible fractional ideal `I` has inverse `1 : I` * Cleanup and documentation * Move `has_div submodule` to algebra_operations * More cleanup and documentation * Explain the `non_zero_divisors R` in the `quotient` section * whitespace Co-Authored-By: Scott Morrison <scott@tqft.net> * `has_inv` instance for `fractional_ideal` * `set.univ.image` -> `set.range` * Fix: `mem_div_iff.mpr` should be `mem_div_iff.mp` (but both reduce to reflexivity anyway) * Add `mem_div_iff_smul_subset` Since that is how the definition of `I / J` is traditionally done, but it is not as convenient to work with, I didn't change the definition but added a lemma stating the two are equivalent * whitespace again (it got broken because I merged changes incorrectly) * Fix unused argument to `inv_nonzero` Co-authored-by: Scott Morrison <scott@tqft.net> Co-authored-by: mergify[bot] <37929162+mergify[bot]@users.noreply.github.com>
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.
This PR defines fractional ideals of an integral domain and proves basic facts about them: they have a
comm_semiringstructure,latticestructure and ideal quotient; the inverse of an ideal, if it exists, is equal to the quotient. These is the work in progress I mentioned on Zulip.The things that are called "fractional ideal of
R" in algebraic number theory are calledfractional_ideal R (non_zero_divisors R)in this file, a notation that probably deserves to be shorter. Do you happen to know a good abbreviation? Alternatively: Is there a good namefooforfractional_idealso that we can abbreviatefractional_ideal R := foo R (non_zero_divisors R)?TO CONTRIBUTORS:
Make sure you have:
If this PR is related to a discussion on Zulip, please include a link in the discussion.
For reviewers: code review check list