In this chapter, we apply Ore's method of localization to study semi-prime Noetherian rings.

The main result of this chapter is Goldie's Theorem (2.3.7). This result is of fundamental importance in the theory of Noetherian rings. The special case of it that is of primary interest to us generalizes (1.1.4) as follows: let R be a semi-prime Noetherian ring, and let CR (0) denote R the set of all regular elements of R. Then CR (0) is an Ore set in R and the Ore localization Q(R) of R at CR (0) is a semi-simple ring.

The usefulness of Goldie's Theorem stems, in part, from its key role in a standard ploy to study semi-prime Noetherian rings. This ploy has led to many useful concepts and patterns of argument. To sketch the ploy, let Q(R) be the Ore localization of a semi-prime Noetherian ring R at CR (0). Since Q(R) is a semi-simple ring, everything about it is nice. Moreover, as seen in chapter 1, the transfer from R to Q(R) takes place in a transparent manner. This raises the possibility that, starting from Q(R) and working backwards, we may be able to glean something nice about R itself. That, in a nutshell, is the ploy.