The king took off his ring - In the ring was the seal of the king. Giving the ring to Mordecai was tantamount to giving him the seal of the kingdom, and constituting him the same as lord chancellor among us.
His ring - i. e. the royal signet by which the decrees of the government were signed.
Over the house of Haman - Not only the building and the furniture, but the household - the vast train of attendants of all kinds that was attached to the residence of a Persian noble.