1 - ORE'S METHOD OF LOCALIZATION

Summary

This chapter is the first of three introductory chapters. These chapters provide the background needed for following the rest of the monograph. For the most part, the material presented in these three chapters is an organization of fairly well-known concepts and results, although the specialist may find a few novelties in the manner in which some of the material is presented.

As stated in the preface, during the last decade and a half, a standard procedure of localization in Noetherian rings has emerged. This standard procedure is based on Ore's method of localization. The first two sections of this chapter are devoted to an examination of Ore's method in its full generality without any reference to the standard procedure of localization. Chapter 2 is devoted to Goldie's Theorem – a highly non-trivial illustration of the usefulness of Ore's method. The standard procedure of localization in Noetherian rings is modelled after the usual procedure of localization in commutative rings and after the way Ore's method is used in Goldie's Theorem. The standard procedure is introduced and studied in chapter 3. Ore's method is used, once again, in chapter 7 to develop a new procedure of localization in Noetherian rings.

Throughout this monograph, we shall primarily be concerned with Noetherian rings which badly fail to be commutative. Experience shows that any straightforward attempt to apply usual commutative methods to these rings leads nowhere. Several fairly common features of these rings seem anomalous in comparison with the features of commutative Noetherian rings.