In this chapter, we examine the structure of an indecomposable injective E over a Noetherian ring R, assuming either that E is tame and R satisfies the second layer condition or that R satisfies the strong second layer condition; in the latter case, E need not be tame.

The focus of our examination is the so-called fundamental series of E. This series is patterned after the idea of ‘layers’ of E and is defined by an inductive use of the idea of the second layer of a module.

It turns out that E is the ascending union of the terms in its fundamental series. Thus, an obvious modification of our earlier explanation of the second layer allows us to view E as a module to be constructed step by step using the successive factors of the terms in its fundamental series. As with the second layer, this then leads to the important, if nettlesome, problem of predicting the composition of the successive factors of the terms in the fundamental series of E. In handling this problem, ‘abstract nonsense’, once again, turns out to be unhelpful. However, we show that an inductive use of the machinery developed in our earlier study of the second layer goes a long way towards making E accessible. Further details can then be obtained by using appropriate special techniques.

In the first two sections, we define the fundamental series of E and study its various aspects. In the third section, we establish the d.c.c. for injective hulls of tame simple modules over polynomial rings and over certain centrally separated rings.