A rolling-element bearing includes an inner ring made of steel and having an inner-ring raceway, an outer ring having an outer ring raceway, and ceramic rolling elements which roll on the inner-ring raceway and on the outer-ring raceway. The inner ring has been subjected to a heat treatment to harden it. The heat treatment is concluded with the performance of a final heat-treatment step at a predetermined temperature. In the inner ring, compressive residual stresses are formed in an outer layer by cold working in the region of the inner-ring raceway. The inner ring also has been subjected to a bluing treatment.