  In this paper we pursue three aims. The first one is to apply Amitsur's
relations and radicals theory to the study of the lattices Id_{A} of closed
two-sided ideals of C*-algebras A. We show that many new and many well-known
results about C*-algebras follow naturally from this approach. To use
"relation-radical" approach, we consider various subclasses of the class A of
all C*-algebras, which we call C*-properties, as they often linked to some
properties of C*-algebras. We consider C*-properties P consisting of CCR- and
of GCR-algebras; of C*-algebras with continuous trace; of real rank zero, AF,
nuclear C*-algebras, etc. Each P defines reflexive relations in all lattices
Id_A. Our second aim is to determine the hierarchy and interconnection between
properties in A. Our third aim is to study the link between the radicals of
relations in the lattices Id_A and the topological radicals on A.
