Nothing first that the class of
all radicals for associative rings forms a lattice under a natural ordering, we show
that several important subclasses, including the class of hereditary radicals, form
sublattices. We give an example showing that the special radicals do not
form a sublattice even though they form a complete lattice under the same
ordering.
After we have studied various lattice-theoretic properties of our main lattices,
showing, in particular, that the lattice of hereditary radicals is Brouwerian, we
determine the atoms of that lattice and show that it has no dual atoms by
computations with free rings. We characterize pseudocomplements in the lattice of
hereditary radicals and give partial results toward determining which radicals of that
lattice are complemented.
