Warm, soapy water is around the limit of what one has to use to clean amethyst. Excessive heat can actually affect the color of amethyst, and sudden temperature changes can cause fractures. Thus, it should not be subjected to a heat-involved cleaning method.
Usually, it is all right to use an ultrasonic cleaner as long as the gemstone has not been fracture-filled. An amethyst ring with diamonds may be subjected to this treatment. If the gemstone is fractured though, the vibrations in the ultrasonic cleaning method can damage the amethyst gem beyond repair.
In fact, it is safer to let the gemstone soak in a safe water-based solution. Fill a bowl with one-quart warm water, and stir in a tablespoon mild dishwashing liquid to it. Then, submerge the amethyst and diamond ring in it. Let the gem soak in it for 15 minutes to let the mild soap dissolve the grime from its surface, and then gently scrub it with a soft and clean bristle toothbrush.
Get rid of the grime and oil residue from it using a soft brush and rinse it thoroughly using distilled water. Then, wipe it gently using a microfiber cloth to absorb the moisture and let it dry before you store it in a box lined with soft fabric.