What is the ruling on a girl not returning the engagement ring after the young man broke off the engagement without explanation, and the mother asked for the ring?

As long as the suitor has not entered into the legal marriage contract, he has the right to demand the return of the gold (the shabaka), and you have no right to any part of it. This is because it is a component of the mahr (dowry) and is not due without the contract. Furthermore, you are not permitted to withhold the shabaka on account of the suitor or his father not contacting your father. However, you may take precautions for yourselves in returning it, such as by having witnesses present or similar measures, to ensure it reaches its owner and to absolve yourselves of his claim to it.