For an arbitrary class of rings M, we have studied, in this paper, some necessary and sufficient conditions for ψM to be closed under homomorphic images or essential extensions, or to be a semisimple class or hereditary class. The main results are: Theorem 4.1 For an arbitrary class of rings M, the following are equivalent: (1) M is a semisimple class; (2) ψM = ψuψM; (3) M~*=(uψM)~*; (4) M^(**)■M~*■(uψM)~*. Theorem 4.3 For an arbitray class of rings M, the follawing are equivalent: (1) ψM is the semisimple class of a hereditary radical; (2) ψM is an essentially closed semisimple class; (3) M~*=(uψM)~* and M~* is essentially hereditary; (4) M~*=(uψM)~* and uψM is essentially hereditary; (5)M~*=(uψM)~* and ψM is essentially closed.
A Class of rings is said to be weakly hereditary if 0≠I△R∈ implies 0≠I^n∈ for some positive integer n, which generalizes the concept of heredity but other than that of regularity. In §2 the properties of such class and its essential cover are studied. In §3 the upper radicals determined by them are investigated. At the same time the 'Problem 42, 44 and 55' of Szasz [7] are discussed. In §4 two examples are given, which show that the concept of weak heredity is independent of that of regularity.
