a rolling bearing has the radius of curvature of the raceway on an inner ring and an outer ring is from not smaller than 50 . 1 % to not greater than 51 . 9 % of the diameter of the balls , and at least one of the compositions of the bearing is obtained by forming an alloy steel having a carbon content of from not smaller than 0 . 50 % to not greater than 0 . 90 %, a chromium content of from not smaller than 3 . 0 % to not greater than 15 . 0 %, a manganese content of from not smaller than 0 . 10 % to not greater than 2 . 0 %, a silicon content of from not smaller than 0 . 10 % to not greater than 2 . 0 %, a molybdenum content of zero or not greater than 2 . 0 % by weight , and a vanadium content of zero or not greater than 2 . 0 %, by weight .