11 - Artinian Quotient Rings

Summary

Goldie's Theorem gives a characterization of those rings which have a classical quotient ring that is semisimple and, in particular, artinian. This naturally gives rise to the question: Which rings have classical quotient rings that are artinian? While this question has a certain abstract interest of its own, its significance turns out to be much greater than one might initially suspect. Rings arising in a natural way frequently have artinian classical quotient rings, and this may be an important fact in their study. In particular, as we shall see in Chapter 14, if R is a subring of a ring S and P is a prime ideal in S, then, while P ∩ R need not be prime or semiprime, it is often possible to show that R/(P ∩ R) has an artinian classical quotient ring.

We first introduce a new notion of rank, known as “reduced rank” (different from the uniform rank introduced in Chapter 5), which is useful in many arguments involving noetherian rings, and we give two naive examples of its use. We then use reduced rank to derive necessary and sufficient conditions for a noetherian ring R to have an artinian classical quotient ring. This basic criterion is very satisfactory in some ways – for instance, it is phrased entirely in terms of properties of individual elements of R – but not in others.