If a property P of ring elements satisfies conditions (a)-(d), then the largest homomorphically closed class having no non-zero P-element is a dual radical in the sense of Andrunakievič, and every dual radical can be obtained in this way. Also properties defined by polynomials are considered and as an application we get various characterizations of the Behrens radical.

Olson and Jenkins defined E(M) to be the class of all rings each nonzero homomorphic image of which contains either a nonzero M-ideal or an essential ideal where M, is any class of rings. E(M) was proven to be a radical class and various classes M were considered. Here the class E(M) is partitioned into two classes: H the class of all rings each nonzero homomorphic image of which has a proper essential ideal and the class H(M) of all rings each nonzero homomorphic image of which contains an M-ideal. It is shown that H is a radical class and under certain conditions H(M) is also a radical class. Various properties placed on M yield several well-known radical classes and an infinite number of supernilpotent nonspecial radical classes is constructed.

For any class of rings it is shown that the class (M) of all rings each nonzero homomorphic image of which contains either a nonzero -ideal or an essential ideal is a radical class. If is a class of simple rings the upper radical generated by , (M), is shown to be equal to (M) where ' is the class of simple rings complementary to .