Engagement jewelry is a category of rings and amulets that is exorbitantly more expensive than regular jewelry. It will raise a pet or hired adventurer's affection if given to them. However, pets prefer to wear engagement jewelry rather than most other equipment, and react badly if one tries to take the jewelry back (they will angrily swallow the item, thus destroying it, and your relationship level will take a big hit). Other than the higher price and affection gains, it acts as regular jewelry. Because of its high value-to-weight ratio, it can be a good thing to trade with other adventurers.