Planetary rings seem to generally form on the ecliptic plane because of the rotation of the planet, but I was wondering what kind of effect the magnetic field of the planet might have on the ring system. Would it help keep the ring system together, disrupt it, or have little to no effect? What if the magnetic poles were on or near the ecliptic?
(I will assume that these planets orbit a star.) Planetary rings are formed by the gravitational capture of small objects, mostly ice and dust. These particles can be magnetic and in such cases would be affected by the planetary magnetic field. It is also to be pointed out that the magnetic field is not really uniform, but let us assume that it is. The presence of a magnetic field would be important for the sustenance of a ring, but the stability of the ring is an entirely different issue. If the ring does consist of magnetically sensitive materials, then large enough fluctuations in the magnetic field can in principle cause the ring to become unstable. So depending on the answer of stability, I choose to say yes, it does depend on the magnetic field. To calculate the stability would be a complex n-body problem but can be averaged over.
There are radial structures in Saturn's rings called spokes, which are likely produced by a mechanism involving electromagnetic effects linked to Saturn's magnetic field. So there would certainly appear to be some aspects of ring phenomena that are affected by magnetic fields.