A radical α in the universal class of associative rings is called matric-extensible if α (Rn) = (α (R))n for any ring R, and natural number n, where Rn denotes the nxn matrix ring with entries from R. We investigate matric-extensibility of the lower radical determined by a simple ring S. This enables us to find necessary and sufficient conditions for the lower radical determined by S to be an atom in the lattice of hereditary matric-extensible radicals. We also show that this lattice has atoms which are
not of this form. We then describe all atoms of the lattice, and show that it is atomic.
We continue our study of the lattice of matric-extensible radicals of associative rings. We find some atoms generated by
simple rings of the lattices of all matric-extensible radicals, matric-extensible supernilpotent radicals and matric-extensible
special radicals. We consider *-rings, which were previously defined by the second author, and consider when they generate
atoms of these lattices.
The base radical class Lb(X), generated by a class X was introduced in . It consists of those rings whose nonzero homomorphic images have nonzero accessible subrings in X. When X is homomorphically closed, Lb(X) is the lower radical class defined by X, but otherwise X may not be contained in Lb(X). We prove that for a hereditary radical class L with semisimple class S(R), Lb(S(R)) is the class of strongly R-semisimple rings if and only if R is supernilpotent or subidempotent. A number of further examples of radical classes of the form Lb(X) are discussed.