Two topics in commutative ring theory

Abstract

This thesis falls into two parts, namely Chapter 1 and Chapter 2, which are completely separate. A brief description of each part follows.

In Chapter 1 we study infinite unions and intersections of cosets of ideals, with a view to obtaining infinite analogues of the finite "prime avoidance" results. In Section 1.1 we examine possible extensions of intersection results from the finite to the infinite. We find that the kind of result that would be useful later in the chapter is not, in general, true. In section 1.2, we establish results that are known so far about ideals which are contained in unions of cosets of ideals. Sections 1.3 and 1.4 generalise these known results, in particular circumstances. In Section 1.5 we describe a "prime avoidance" property that we should like a set of ideals to have, and examine the implications of this property. Section 1.6 uses the results of Sections 1.3 and 1.4 to give examples of the behaviour described in Section 1.5. Sections 1.7 to 1.11 are concerned with applications of the results obtained earlier. In Section 1.7 we give infinite analogues of the finite "prime avoidance" results. In Sections 1.8 and 1.9 we generalise standard results from (Kap] on zero-divisors and regular sequences. In Section 1.10 we apply the results of Section 1.9 to big Cohen-Macaulay modules.

In Chapter 2 we develop the idea of Zariski regularity to prove a uniform Artin-Rees theorem and to generalise the Main Lemma of [Zar]. We begin by defining and characterising the notion of Zariski regularity, in Section 2.1. We show, in Section 2.2, that Zariski regularity is an open condition and move on to prove the uniform Artin-Rees theorem, in Section 2.3. Finally, in Section 2.4, we generalise Zariski's Main Lemma to a wider class of rings and to modules over these rings.