By using class operators, we define base radical and semisimple classes, within a broad abstract setting due to Puczylowski. The new notions agree with the usual Kurosh–Amitsur ones for associative rings and groups but differ for not necessarily associative rings, in general lying strictly between the Kurosh–Amitsur and torsion theory notions. A study of the class operators for their own sake is initiated, and a connection with modal logic is made.