What is the ruling on marrying another woman while keeping the current wife under one's marital authority, given that she is negligent in religious matters and does not care for the house, and sixteen years of advising her have passed without success, along with the husband's desire not to wrong her?

Allah has granted the husband the right of qawamah (guardianship/headship) over his wife, which means he is to ensure she performs her obligatory duties and abstains from prohibitions. The husband must teach his wife and urge her to obey Allah. He can do this by having her listen to admonitions, introducing her to righteous Muslim women, and supplicating for her. If the wife does not respond to her husband's guidance in religious matters, it is preferable for him to divorce her. It is also the husband's right to marry another woman if he is capable of being just between the two wives.