What is the ruling on a wife agreeing to relinquish a quarter of the house she owns to her deceased husband's brother, in exchange for her living in it for the rest of her life, provided that her heirs have no share in it after her death?

It is not permissible for a woman to waive her share [of inheritance] after her death in exchange for her usufruct of three-quarters of the house for the duration of her life, because this is a void ijarah (lease/rental contract) due to the unknown duration. An ijarah must have a known duration. To rectify this transaction, it is necessary to specify the duration of the ijarah, even if it is long, with the payment of the rent (which is one-quarter of the house) at the end of the period. If the woman dies before then, her heirs will benefit from the house until the end of the period, after which the brother will take it. If the period expires and the woman is still alive, she is obligated to surrender her share.