摘要
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.
