gh-37719: Refactor ring categories
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
### :memo: Checklist
- [x] The title is concise and informative.
- [x] 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
URL: https://github.com/sagemath/sage/pull/37719
Reported by: Frédéric Chapoton
Reviewer(s): Matthias Köppe, Travis Scrimshaw
