You just know that Agnes’s ring and locket are going to be important because so much mystery is associated with them. When Agnes first shows up at the workhouse to give birth to Oliver before dying, one of the very few things we learn about her is that she wasn’t wearing a wedding ring. The question of who she was, and whether there were ever a wedding ring at all, is always hanging around in the back of everyone’s mind.
The first piece of the puzzle we pick up is that old Sally had stolen some kind of jewelry from Oliver’s dying mother the night that he was born – and that it was something that contained a clue as to his parentage and identity. This gold jewelry, we finally learn in Book III, Chapter One, included a gold locket with two locks of hair, and a wedding ring. The locket represents the physical union between Agnes Fleming and Edward Leeford, Oliver’s father – it contains a lock of each of their hair, physically bound together. But a locket is designed to be "locked" and kept a secret. The kind of union it represents isn’t the kind of union that the world recognizes. The ring would represent a union that the world would recognize, but the ring was never completed: it has Agnes’s name carved into it, but only her first name. Her maiden name, "Fleming," was given up when she lost her, um, maidenhood; and she never took the next step to become "Agnes Leeford" by marrying Edward. So the ring, which should (and usually does) represent unity, in Oliver Twist, represents only an incomplete union.