The interest in the second layer of a tame indecomposable injective is, hopefully, clear at this stage. The next thing to do is to investigate the tractability and the usefulness of the second layer. This turns out to be a rather elaborate and convoluted affair, with a variety of techniques having some bearing on various aspects of it. For the sake of clarity of exposition, we have divided it, roughly, into three parts: development of basic tools, study of the tractability of the second layer, and applications to localization. The first of these three, development of basic tools, is the focus of this chapter.

To investigate the second layer, the main tool we employ consists of links between prime ideals of a Noetherian ring. The importance of links stems from three facts: first, links are constructed from nothing but the ideal structure of the ring and, thus, are amenable at least when the ring is reasonably well-behaved. Secondly, links virtually determine the tame part of the second layer; cf., (6.1.3). Thirdly, links arise in connection with an important property of Ore sets; cf., (5.4.5).

The other major tool we employ consists of Noetherian bimodules. These bimodules, and their peculiarities, are of interest in their own right. Moreover, they provide a mechanism which allows one to exploit, simultaneously both sides of a Noetherian ring. Such a simultaneous use of both sides is often crucial in obtaining deeper properties of Noetherian rings. Noetherian bimodules of a special type, called bonds, are intimately related to links, and are often needed to strengthen, complement, and utilize the information provided by links.