sagemathgh-36917: `sage.rings`: Modularization fixes
    
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Standard modularization changes:
- Using more lazy imports, more specific imports and similar
- Some try... except for imports of element implementations
- `# needs`
- incidental doctest cosmetics

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This is:
- Cherry-picked from sagemath#35095
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### ⌛ Dependencies

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URL: sagemath#36917
Reported by: Matthias Köppe
Reviewer(s): David Coudert

<!-- ^^^^^
Please provide a concise, informative and self-explanatory title.
Don't put issue numbers in there, do this in the PR body below.
For example, instead of "Fixes sagemath#1234" use "Introduce new method to
calculate 1+1"
-->
<!-- Describe your changes here in detail -->
Standard modularization changes:
- Using more lazy imports, more specific imports and similar
- Some try... except for imports of element implementations
- `# needs`
- incidental doctest cosmetics

<!-- Why is this change required? What problem does it solve? -->
<!-- If this PR resolves an open issue, please link to it here. For
example "Fixes sagemath#12345". -->
This is:
- Cherry-picked from sagemath#35095
<!-- If your change requires a documentation PR, please link it
appropriately. -->

### 📝 Checklist

<!-- Put an `x` in all the boxes that apply. -->
<!-- If your change requires a documentation PR, please link it
appropriately -->
<!-- If you're unsure about any of these, don't hesitate to ask. We're
here to help! -->
<!-- Feel free to remove irrelevant items. -->

- [x] The title is concise, informative, and self-explanatory.
- [ ] 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.

### ⌛ Dependencies

<!-- List all open PRs that this PR logically depends on
- sagemath#12345: short description why this is a dependency
- sagemath#34567: ...
-->

<!-- If you're unsure about any of these, don't hesitate to ask. We're
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URL: sagemath#36917
Reported by: Matthias Köppe
Reviewer(s): David Coudert