Are the following phrases considered a divorce: "I divorce you by three, and your mother is forbidden to me, that I supplicate for you in my prostration," and "Indeed, the problems between me and you lead to problems between me and your mother, and you will be the cause of your mother's divorce"?

The question is ambiguous, but if a triple divorce and the prohibition of the wife were made contingent upon the son's supplication during prostration, then if he supplicated, the divorce would occur according to the majority of scholars. According to Ibn Taymiyyah, an expiation for an oath (kaffarat yamin) is due if he did not intend divorce. If the divorce was made contingent upon the absence of supplication, it would occur if the supplication did not take place. If the oath was false, the divorce is binding according to the majority of scholars.

As for "your mother is forbidden (haram)," it depends on the speaker's intention: if he intended divorce, it is a divorce; if he intended ẓihār (a form of injurious oath likened to calling one's wife one's mother), it is ẓihār; and if he intended an oath, it requires an expiation for an oath. Ẓihār occurs along with prohibition if their cause occurs at the same time. The expiation for ẓihār is not waived unless he marries her after she becomes lawful to him after marrying another man and then being divorced by him after consummation.

The statement by the man that the problems between him and his son are a reason for divorcing the wife does not result in a divorce; rather, it is a statement of information. The son must honor his father and obey him in matters that do not involve disobedience to Allah, and he should not be a cause of separation between his parents.