An inner ring or and outer ring for a roller bearing includes: 0.30-0.45 wt. % carbon, 0.1-0.7 wt. % silicon, 0.6-0.9 wt. % manganese, 0.9-1.2 wt. % chromium, 0.15-0.7 wt. % molybdenum, 0-2.0 wt. % nickel, 0-0.02 wt. % phosphorus, and 0-0.02 wt. % sulfur, the balance being iron and unavoidable impurities. The microstructure of the steel composition contains bainite, and a carbonitrided case layer is provided on a surface of the inner ring or outer ring.