The air-conditioning compressor is provided with a casing 3, with a shaft 4 able to rotate about an axis 5, and with a torque-transmitting device 1 comprising a pulley 7, a rolling bearing 8 positioned inside the pulley 7 and comprising an inner ring 14, an outer ring 15 and at least one row of rolling elements 16, and a torque-transmitting member 9 equipped with an external part angularly connected to the pulley 7, with an internal part 20 fixed to the shaft 4 and with at least two arms 21 connecting the external part and the internal part. Empty spaces 23 are formed between the arms 21, the bore 14b of the inner ring 14 having a radius greater than the distance between the axis 5 of rotation and an internal end 23a of the said empty spaces 23 so as to allow the insertion of a crimping tool. Local plastic deformations 30 of the casing interfere with a transverse face 14c of the inner ring 14, the said local plastic deformations 30 being positioned in axial alignment with the said empty spaces 23 in at least one relative angular position between the torque-transmitting member 9 and the inner ring 14.