What is the ruling on a husband telling his wife, "You are not under my responsibility," without intending divorce, and what is the ruling on an oath of "by divorce" uttered unintentionally?

It must be noted that a wife is not permitted to request a divorce without a legitimate Shar'i (Islamic legal) excuse, due to the Prophet's (peace be upon him) saying: "Any woman who asks her husband for divorce without a sound reason, the scent of Paradise will be forbidden to her." The phrase "not under my responsibility" (مش على ذمتي) is an idiomatic expression (kinayah), and divorce does not occur by it unless one intends thereby to effect a divorce. Since you did not intend that, there is nothing against you. Sheikh Ibn Baz (may Allah have mercy on him) was asked about the ruling on someone who swore an oath of divorce inadvertently, without realizing it, and he replied that there is nothing upon him. Therefore, if you uttered the divorce without intention, nothing is incumbent upon it, and your wife remains under your marital bond.