sage.rings: Modularization changes

<div id="comment:0"></div>

split out from #32432.



CC:  @kliem

Component: **refactoring**

Author: **Matthias Koeppe, Jonathan Kliem**

Branch/Commit: **[`92d88c0`](https://github.com/sagemath/sagetrac-mirror/commit/92d88c059f0608e19e951e30f1c0afe2d2bb38f6)**

Reviewer: **Jonathan Kliem, Matthias Koeppe**

_Issue created by migration from https://trac.sagemath.org/ticket/32870_

