Gradations of finite rings and homogeneity of their radicals

Any gradation of a finite field by a cancellative monoid is equivalent to a strong gradation by a finite cyclic group. A formula for the number of gradations of finite fields by cancellative monoids, up to equivalence, is given. The prime power factors of finite rings are shown to be homogeneous. The homogeneity of the nil, Jacobson, and prime radicals of any finite ring is completely determined by the homogeneity of the corresponding radicals of its prime power factors.