Refactor ring categories

[fchapoton]

A somewhat large refactoring of some of the auld classes for rings, and related categories

• introducing a new category of Noetherian rings
• moving some methods in appropriate categories
• fixing necessary details

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

Depends on #37778

[fchapoton]

damn, there remains annoying failing doctests in src/sage/matrix !

[fchapoton]

One of the remaining bugs is now

sage: A = Zmod(1)
sage: B = A['x']
sage: B.one()
crash in coercion somehow

[tscrim]

Okay, I've figured it out. The change in the category from Algebras to Rings means that the base ring coercion is no longer provided by default. It's actually somewhat amazing that this doesn't end up going through an infinite loop, but that is how coerce_map_from works with its caching. If you add

        if P == base_ring:
            return self._coerce_map_from_base_ring()

to the beginning of PolynomialRing_general._coerce_map_from_(), then the coercion will work. It's a bit of a dirty workaround, but I don't see how to do it any better. The other option I considered is doing a second base_ring.is_zero() at the end of the __init__ and registering the coercion manually. Yet this felt even less clean, but I can't really say why though. (I am also surprised the multivariate case works; I don't understand how it does at this point.)

There are some other issues/inconsistencies with polynomials over $F_1$:

sage: B = A['x,y']
sage: B.gens()
(0, 0)
sage: B = A['x']
sage: B.gens()
(x,)

The univariate case is wrong IMO.

The change in multi_polynomial.pyx is for the polynomial ring with no generators, correct? Can we add a test for this?

I am not sure about the change for LocalGeneric to Parent from CommutativeRing because the latter class seems to have methods not provided by the category (e.g., localization).

[fchapoton]

Thanks a lot, Travis ! I was worrying to be unable to debug that issue about Zmod(1)[x].

I propose to keep the other inconsistencies of polynomials over the zero ring for later.

I have added the suggested doctest in multi_polynomial.

For the change in local_generic, I do not remember why I was forced to do that. In any case, it contributes to the future possible removal of CommutativeRing, one day.

I have disabled a doctest in the testsuite of Laurent polynomials over Zmod(2) about gcd.

[fchapoton]

no idea what the CI complains about. Something about "modularity", it seems......

[mkoeppe]

no idea what the CI complains about

The failure is unrelated to this PR and tracked in #37734.

[mkoeppe]

LGTM

[grhkm21]

Setting to positive since Matthias approved, unless Travis disagrees.

[tscrim]

Thank you. Once the bot comes back green, positive review.

[github-actions]

Documentation preview for this PR (built with commit eb758e6; changes) is ready! 🎉
This preview will update shortly after each push to this PR.

[fchapoton]

As expected, the lights are not green. As I said, I have no idea if it is possible to declare a dependency in such a way that the CI will merge the branches before testing.

[mantepse]

As expected, the lights are not green. As I said, I have no idea if it is possible to declare a dependency in such a way that the CI will merge the branches before testing.

As far as I know, you have to merge the dependency into your branch.

[mkoeppe]

I've repaired the CI. It complains:

sage -t --random-seed=232623843441133993949100432847335930207 src/sage/matrix/matrix_space.py
**********************************************************************
File "src/sage/matrix/matrix_space.py", line 493, in sage.matrix.matrix_space.MatrixSpace
Failed example:
    MatrixSpace(ZZ, 10, 5).category()
Expected:
    Category of infinite enumerated finite dimensional modules with basis over
     (Dedekind domains and euclidean domains
      and infinite enumerated sets and metric spaces)
Got:
    Category of infinite enumerated finite dimensional modules with basis over (Dedekind domains and euclidean domains and noetherian rings and infinite enumerated sets and metric spaces)
**********************************************************************
File "src/sage/matrix/matrix_space.py", line 497, in sage.matrix.matrix_space.MatrixSpace
Failed example:
    MatrixSpace(ZZ, 10, 10).category()
Expected:
    Category of infinite enumerated finite dimensional algebras with basis over
     (Dedekind domains and euclidean domains
      and infinite enumerated sets and metric spaces)
Got:
    Category of infinite enumerated finite dimensional algebras with basis over (Dedekind domains and euclidean domains and noetherian rings and infinite enumerated sets and metric spaces)
**********************************************************************
1 item had failures:
   2 of  50 in sage.matrix.matrix_space.MatrixSpace
    [4[60](https://github.com/sagemath/sage/actions/runs/8948624007/job/24713068325?pr=37719#step:12:61) tests, 2 failures, 2.97 s]

[mkoeppe]

You can fix the failure in "test-mod" like this:

diff --git a/pkgs/sagemath-categories/known-test-failures.json b/pkgs/sagemath-categories/known-test-failures.json
index ab27f65ca8d..f8303460c94 100644
--- a/pkgs/sagemath-categories/known-test-failures.json
+++ b/pkgs/sagemath-categories/known-test-failures.json
@@ -597,6 +597,10 @@
         "failed": true,
         "ntests": 126
     },
+    "sage.categories.noetherian_rings": {
+        "failed": true,
+        "ntests": 19
+    },
     "sage.categories.number_fields": {
         "failed": true,
         "ntests": 41

(see https://deploy-livedoc--sagemath.netlify.app/html/en/developer/doctesting#updating-baseline-files)

[mkoeppe]

check whether you want to use "Noetherian" or "noetherian" throughout the text

[mkoeppe]

LGTM.