What is the ruling on a wife buying a new living room set without her husband's permission, even though he swore not to buy another?

Your breaking of the oath depends on your intention. If you intended never to buy the set, or not to buy it as long as you are abroad, then you have broken your oath and expiation is required. However, if your intention was not to buy it yourself, or with your permission, or with your money, and so on, then you have not broken your oath and nothing is required of you. Your wife erred by disobeying you and disposing of your money without permission, and she should repent for that. We advise you to forgive and pardon her.