At present, four classes of Noetherian rings are regarded as reasonably well-understood. These are the classes of hereditary Noetherian prime rings with enough invertible ideals, Noetherian P.I. rings, enveloping algebras of solvable Lie algebras, and group rings of polycyclic-by-finite groups over commutative Noetherian rings. The rings in these four classe are well-behaved and are often cited as natural and important instances in which the theory of Noetherian rings, in its present form, is useful. Indeed, parts of the present theory were developed with an eye towards these rings.

The classes mentioned above are usually studied one at a time. Substantial amount of information is known about each one of them. It is also known that these classes do not have non-trivial overlaps although they do have some vaguely analogous features.

Here, we deal with the four classes stated above. Our primary aim is to provide some information about them which is related to the material in the monograph. To this end, we show, first, that the rings in these classes satisfy the strong second layer condition. This provides an abstract unification of these classes and accounts for their vaguely analogous features. Secondly, we show that, at least for enveloping algebras and for group rings, infinite cliques can hardly be regarded as exceptional since their existence is another way of expressing the distinction between mere solvability and nilpotency. The import of this for localization has been discussed in the Preface and in chapter 7.