What is the ruling on a husband saying to his wife, "Divorce me if you don't want me," whereupon she says to him, "You are divorced," knowing that divorce is not within her authority, and does the divorce take effect in this case?

If the husband says to his wife, "Divorce me," and he does not intend to delegate the divorce to her, or he does intend it but the wife does not intend divorce by her saying, "You are divorced," then no divorce occurs. One must not be lenient in uttering words of divorce, as it may harm the marital relationship.