A radical classRpartitions a class of ringsTif for every T ε T either R(T) = 0 orR(T) = T. Given any class of rings C, Stewart [8] defines the class of almostC-rings, A(C), to be all those rings every non identity homomorphic image of which is inC. He then defines C to be the class of all rings S where S is the heart of a ring A with A/S in C. He then considers the general question: what radical classes partition the class of almost C-rings where C has certain properties. With the aid of M*-rings introduced in this paper we can generalize some of Stewart's results and obtain more natural partitions.