Jamieson, Fausset and Brown on 1 Kings 2:19-20

19-20. Bath-sheba ... went unto King Solomon--The filial reverence and the particular act of respect, which Solomon rendered, were quite in accordance with the sentiments and customs of the East. The right hand is the place of honor; and as it expressly said to have been assigned to "the king's mother," it is necessary to remark that, when a husband dies, his widow acquires a higher dignity and power, as a mother over her son, than she ever possessed before. Besides, the dignity of "king's mother" is a state office, to which certain revenues are attached. The holder has a separate palace or court, as well as possesses great influence in public affairs; and as the dignity is held for life, it sometimes happens, in consequence of deaths, that the person enjoying it may not be related to the reigning sovereign by natural maternity. Bath-sheba had evidently been invested with this honorable office.