In this chapter, we present the standard procedure of localization in Noetherian rings as it is developed in the literature. The strengths and the limitations of this procedure have been commented upon in the Preface.

We describe the considerations that guided the development of this procedure. After Goldie's Theorem, there remained little doubt as to the usefulness of Ore's method of localization. Thus, in view of the fact that localization at prime ideals is repeatedly used in the theory of commutative Noetherian rings, it seemed natural and promising to make an attempt to use Ore's method to ‘localize’ Noetherian rings at prime ideals.

The first such attempt was made by Goldie (67),(68). Goldie's attempt was instrumental in crystallizing the ideas involved and led to the notions of localizable prime ideals and classically localizable prime ideals. Both of these notions use the set CR (P) to localize a Noetherian ring R at a prime ideal P. Some cogent reasons for shifting attention from prime ideals to at least semi-prime ideals were first supplied by Jategaonkar (73a), (74b).

As stated in the Preface, classically localizable semi-prime ideals are usually regarded as the main objects in the study of localization in Noetherian rings. In the Preface, to keep the matters simple, we refrained from indicating that the apparently more general localizable semi-prime ideals are also considered worthy of attention. Semi-prime ideals of both of these types are studied in this chapter. Our results show, roughly, that these are just the semi-prime ideals at which ‘localization’ is sufficiently well-behaved to be usable.