Localisation in P.I. Rings

In order to localise with respect to some multiplicative set S in a noncommutative ring, certain conditions must be satisfied by S. When such conditions are satisfied, the resulting localisation is either a left or right localistion (as will be explained) but, in general, not both. The main aim of this paper is to show that in the case where R is a prime Noetherian P.I. ring, all localisations are in fact two sided and can be thought of as a localisation with respect to a certain collection of prime ideals of R. The project is split into seven chapters which we outline below. Chapter 1 Here we introduce P.I. rings and outline some of their basic properties. Chapter 2 In this chapter we state and prove two important theorems regarding P.I. rings. These are Kapansky’s Theorem and Posner’s Theorem. Chapter 3 Using Posner’s Theorem, we introduce the Trace Ring of a prime P.I. ring R. We show that, in this extension ring, localisation is particularly wellbehaved. Chapter 4 Here we introduce, more formally, the idea of localisation in a noncommutative ring. We also introduce the idea of links between prime ideals. Chapter 5 In this chapter a new class of rings, called F.B.N. rings, are introduced. It will be shown that Noetherian P.I. rings belong to this class. Chapter 6 Here we develop the idea of Krull/Classical Krull dimensions and show that for F.B.N rings these dimensions coincide. Chapter 7 Finally, in this chapter we tackle the proof of our main theorem. A description of the link structure of a prime Noetherian P.I. ring is also given.