Now that we have at our disposal some potent weapons, primarily the main Lemma, we return to the problem of using Ore's method to develop workable machinery of localization in Noetherian rings.

A hint as to how such machinery can be developed is offered by the Stability Theorem. Simply put, it says that, so far as the use of Ore's method is concerned, a clique of prime ideals should be regarded as a single entity. Since cliques can, and often do, contain infinitely many prime ideals, this leads one to consider ‘classical localization’ at possibly infinite sets of prime ideals.

The study of such possibly infinite ‘classical sets’ of prime ideals is the focus of this chapter.

Our main result on classical sets is the characterization (7.1.5). For cliques, studied in the second section, this characterization becomes simpler, and leads to some useful sufficient conditions for classicality. The upshot is that, in several types of important Noetherian rings, one can localize at any clique without a hitch. Some early applications of this machinery are promising. Further applications are needed to clarify its scope.

Classically localizable semi-prime ideals are studied, once again, in the third section. In (7.3.1), a convenient characterization of such semi-prime ideals is proved. It reveals the ‘real’ reason for the stymie stated in the Preface. It also leads to several useful sufficient conditions for a semi-prime ideal in a Noetherian ring to be classically localizable.

Classical localization at the nil radical is used in the fourth section to provide a coherent approach to Noetherian orders in Artinian rings.