Title:

Normal elements and prime ideals in Noetherian rings

This thesis consists of three chapters, which are loosely linked by the concept of normal elements (an clement x of a ring R is normal if xR = R.\). In Chapter 1 we examine a class of rings known as Noetherian unique factorisation rings (Noetherian UFRs). These arc prime Noetherian rings in which every prime ideal of height one is generated by a normal element. The main result of this chapter is that if A is a commutative UFD and G is a polycyclicbyfinite group such that the group ring AC is a Noetherian UFR then the set C consisting of the elements of K which are regular modulo all of the height one prime ideals of R is an Ore set in R; we also describe some aspects of the structure of the rings obtained by localising AC at the set C. In Chapter 2 we study the process of localisation at prime ideals which are generated by a regular normalising sequence. Section 2.1 contains a summary of the theory of localisation at cliques in Noetherian rings; in 2.2 we prove some useful results related to this theory. In section 2.3 we study the properties of regular normalising sequences and identify the clique of a poly(regular normal) prime ideal (i.e., a prime generated by a regular normalising sequence). In section 2.4 we define a class of rings called RL rings, which are the rings obtained by localising at the clique of a poly(rcgular normal) prime ideal. We describe various properties of these rings: in particular, we calculate the classical Krull dimension and the global dimension of such a ring (T, say), and show that they are both equal to the length of any regular normalising sequence generating any maxima] ideal of T. In section 2.5, we apply homological methods to RL rings to study the heights of prime ideals. The main result here is a height formula, which states that if g is a finitedimensional complex soluble Lie algebra and T is the ring obtained by localising at the clique of any prime ideal in U(g), the universal enveloping algebra of g, then we have ht(Q) + Kdim(T/Q; = Kdim (T) for every prime ideal Q of T. In Chapter 3, we consider a ring of skew Laurent polynomials T = R[S 1 ..... S jfl; a i ..... a „] with a j (. Aut R. In 3.1 and 3.2 a map of Laurent is constructed from T to the ring 5 = R[x ^ polynomials in n central indcterminates x\ .... xn and it is shown that if R is commutative then this map induces an isomorphism between the lattice of twosided ideals of T and the lattice of Gstable ideals of S, where G is the subgroup of Aut 5 generated by 01, ..., on; furthermore, under this isomorphism the prime ideals of 7" correspond to the Fprime ideals of 5, where F 3 G is a certain subset of End 5. In section 3.3 we show that with appropriate restrictions on R and G the ring T is normally separated; this yields information about the height of prime ideals in 7" and S, and enables us to prove in 3.4 that the ring T is often catenary.
