The amethyst ring that Father bought for Mother when he was in America long ago is a symbol of their love for each other. And since Father has been M.I.A. for nine years and Mother still wears the ring, it's also a symbol of Mother's steadfast devotion to Father and his memory. When Mother considers selling the ring in order to buy things they need, her sons convince her that she shouldn't, because they believe it is "… the last/ tangible remnant of love"(2.15.4). And the thing is, they're pretty much right.
The ring isn't just something for Mother to feel connected to Father with, though, and Hà depends on it as well, which we know since she has to touch it every night before going to sleep. In this way, it symbolizes stability and routine, and clues us into the fact that Father counts in the family's daily lives even if he hasn't been around for nearly a decade.
When Mother loses the ring at the factory she works in, and the family searches for it but comes up empty, Mother—who has been waiting for a sign regarding Father and his plight—interprets this as a message that Father is dead and never going to return. So just as the ring represents Mother and Father's relationship, the loss of the ring formalizes the end of their time together, even though Father hasn't been on the scene for nine years at this point. When this happens, Mother lets go of Father, and the family performs a ritual to say goodbye to him.