On the lattice of matric-extensible radicals

A radical α in the universal class of associative rings is called matric-extensible if α (R _n) = (α (R))_n for any ring R, and natural number n, where R _n 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.