Is it permissible in Islamic law for a wife to remain married to her husband solely as the mother of their children, without marital intimacy, given her reluctance to divorce for the sake of the children, and after her husband neglected her, married another woman, and traveled with her?

It is forbidden for a husband to neglect one of his wives and turn away from her; he will come on the Day of Resurrection with his side inclined. Similarly, it is forbidden for a wife to refuse to fulfill her husband's conjugal rights and other entitlements.

A wife should not seek divorce merely because her husband has married another, unless this is accompanied by clear and irremediable harm. If there is no harm, she may, by her own consent, waive some of her rights and remain under his marital bond, in accordance with the Almighty's saying: "And if a woman fears from her husband high-handedness or desertion, there is no sin upon them if they make terms of settlement between themselves."