What is the ruling of Islamic law regarding a mother threatening her daughter with her displeasure and refusal to forgive her if she does not relinquish her share of the inheritance, which the mother claims to own without proof? And would the daughter be considered sinful or disobedient to her mother if she refused to give up her right?

The fundamental principle is that a person's ownership transfers to their heirs after their death, and it is not permissible for anyone to take an heir's share or demand that they relinquish it without their full consent, based on the words of the Almighty: "For men is a share of what the parents and close relatives left, and for women is a share of what the parents and close relatives left—from what is little or much—a share [of inheritance] ordained," and according to the Prophet, peace and blessings be upon him: "The wealth of a Muslim is not lawful except with his consent."

Therefore, if your wife's mother does not have proof of her ownership of a portion of your wife's share, your wife is not obligated to relinquish it, and her not doing so is not considered an act of disobedience. The wife should be gentle in her dealings with her mother and involve people of goodwill. If she willingly relinquishes a part of her share to her mother, seeking reward, then that is good, but it is not obligatory.