Is a man's statement to his mother-in-law, "Your daughter is divorced," considered a first divorce for his wife, and what is the method for him to take his wife back?

It is not permissible for a wife to leave her husband's home without a legitimate Shar'i (Islamic legal) excuse. Doing so is considered nushuz (disobedience/rebellion). Her dispute with her husband's mother does not justify her leaving. The behavior of the wife's family towards the husband's mother constitutes harm to a Muslim. The wife has a right upon her husband to be provided with an independent dwelling. The husband's statement to his wife's mother, "Your daughter is divorced," is considered a single, revocable divorce (talaq raj'i). He can take her back (raj'a) before the 'iddah (waiting period) ends, provided this divorce does not complete the three divorces. The retraction is effected by his saying: "I have taken my wife back," or "I have restored her to my 'ismah (bond of marriage)," or similar phrases, without requiring the wife's presence, knowledge, or consent.