The statement that the router IDs of OSPF routers in the same area must be the same is incorrect. In OSPF, each router must have a unique router ID within the entire OSPF network, not just within an area. The router ID can either be manually configured or automatically assigned from the highest IP address on an active router interface. The router ID does not have to be the IP address of an OSPF router interface, though it often is if not manually configured. Lastly, a router running OSPF does need to have a router ID for it to function properly.