What is the ruling on beating one's wife, and are there any cases in which beating is permissible? What are the Qur'anic verses and Prophetic traditions related to this?

Allah has forbidden injustice and aggression. Striking one's wife without a legitimate religious justification is an act of aggression and injustice that is impermissible, for it harms her and violates Allah's command to treat one's spouse with kindness.

However, a non-violent beating of the wife is permissible—with conditions and regulations—if she has become rebellious (nashazat) and disobedient to her husband after he has admonished her and forsaken her in bed, based on the Almighty's saying: "As for those [wives] from whom you fear defiance, admonish them, and abandon them in bed, and strike them. But if they obey you, do not seek a way [of harm] against them."

Conditions and Regulations for Striking: It must not be violent (mubarreh). He must not strike her on her face. He must not insult her with degrading terms. The intention behind it must be to reform the wife and encourage her obedience, not revenge. He must desist from it once the objective is achieved.

Both spouses should observe mercy, overlook mistakes, and be guided by the Prophetic tradition to avoid reaching this stage.