Closure properties in radical theory

On hereditary radicals

“…Hereditary radicals appear usually by lower radical constructions. A~ME~DA~IZ [2] and RJABUtIIN [7] Observe that condition (i) is not necessary to the hereditariness of the upper radical. Consider namely the class of all primitive rings in the sense of J~co~so~.…”

Lattices of radicals

“…Also, a(R n ) = I n for some ideal I of R by LEMMA 7. Therefore, a((R/I) n ) = a{RJI n ) = 0, which implies a(R/I) -0 by (3). This implies that 7 3 a(R).…”

Strong heredity in radical classes

“…Also as before (induction) ^£' £ Sf where Stf is any homomorphically closed hereditary class with property (a) containing (1), (2), (3), and (4) go through as in the associative case (see [4]) using Theorem 2.1 and Lemma 1.3. Conversely, suppose ^ satisfies (1), (2) so iΓ is an ideal of R containing ^(T), and if is proper in T, a contradiction which proves the theorem.…”