This chapter is the first of two chapters in which we study Noetherian rings that satisfy the second layer condition.

The importance of the second layer condition for localization is clear from the results in chapter 7. In the next two chapters, we show that the second layer condition is of importance for diverse other reasons as well. On the whole, the results in these two chapters as well as those in chapter 7 show that Noetherian rings satisfying the second layer condition are accessible, pleasant, and useful. Based on these results, it would not be too inaccurate to maintain that Noetherian rings which are wellunderstood at present from a general point of view are precisely those that satisfy the second layer condition.

A comment on the dichotomy based upon the second layer condition may be helpful. The techniques employed in this chapter as well as in the next chapter rely heavily on Noetherian bimodules and links. These techniques, and their consequences, bring out the sharp differences between Noetherian rings that satisfy the second layer condition and Noetherian rings that do not satisfy the second layer condition. The study of the latter type of rings awaits different techniques.

In this chapter, we concentrate on the ring theoretic implications of the second layer condition. In chapter 9, we examine the import of the second layer condition for the structure of indecomposable injectives. In the Appendix, we show that certain ‘natural’ Noetherian rings satisfy the second layer condition.