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The theory of Noetherian rings may be said to have begun with Goldie's 1958 paper. Since then the theory has steadily gathered strength. By now, various methods and results from the theory of commutative Noetherian rings have been adapted to non-commutative Noetherian rings. The theory has also contributed to and benefitted from developments in three nearby areas: P.I. rings, enveloping algebras of Lie algebras, and group rings of polycyclic-by-finite groups.
A major problem still confronting the theory of Noetherian rings concerns localization. To get an idea of this problem, recall that several fundamental results in the theory of commutative Noetherian rings are obtained by using the procedure of localization at prime ideals. It is then natural to expect – at least to hope – that a similar procedure which enables one to localize Noetherian rings at prime ideals would have a similar salutary effect on the theory of Noetherian rings. The problem, in a nutshell, is to find such a procedure.
During the last decade and half, quite a bit of effort has been made to find such a procedure. By now, a certain procedure has emerged as the ‘correct’ one. This standard procedure takes the commutative situation and the situation dealt with in Goldie's Theorem (2.3.7) as models, and attempts to use Ore's method of localization to localize Noetherian rings at semi-prime ideals. The general belief as to the ‘correctness’ of the standard procedure is based on some internal evidence along with some guesswork.
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