The ring: place your ring in the resealable foil pouch it came in, seal it, and dispose in your household garbage (outside the reach of pets and children).
The patch: fold the sticky parts of your patch together, place in an airtight container (like a resealable bag), and dispose in your household garbage (outside the reach of pets and children).
Never flush the ring or patch down the toilet, as the hormones can leach into our groundwater supply.