Title:

Noncommutative Noetherian Unique Factorisation Domains

The commutative theory of Unique Factorisation Domains (UFDs) is welldeveloped (see, for example, ZariskiSamuel[75], Chapter 1, and Cohn[2l], Chapter 11). This thesis is concerned with classes of noncommutative Noetherian rings which are generalisations of the commutative idea of UFD. We may characterise commutative Unique Factorisation Domains amongst commutative domains as those whose heightl prime ideals P are all principal (and completely prime ie R/P is a domain). In Chatters[l3], A.W.Chatters proposed to extend this definition to noncommutative Noetherian domains by the simple expedient of deleting the word commutative from the above. In Section 2.1 we describe the definition and some of the basic theory of Noetherian UFDs, and in Sections 2.2, 2.3, and 2.4 demonstrate that large classes of naturally occuring Noetherian rings are in fact Noetherian UFDs under this definition. Chapter 3 develops some of the more surprising consequnces of the theory by indicating that if a Noetherian UFD is not commutative then it has much better properties than if it were. All the work, unless otherwise indicated, of this Chapter is original and the main result of Section 3.1 appears GilchristSmith[30]. In the consideration of Unique Factorisation Domains the set C of elements of a UFD R which are regular modulo all the heightl prime ideals of R plays a crucial role, akin to that of the set of units in a commutative ring. The main motivation of Chapter 4 has been to generalise the commutative principal ideal theorem to noncommutative rings and so to enable us to draw conclusions about the set C. We develop this idea mainly in relation to two classes of prime Noetherian rings namely PI rings and bounded maximal orders. Chapter 5 then returns to the theme of unique factorisation to consider firstly a more general notion to that of UFD, namely that of Unique Factorisation Ring (UFR) first proposed by ChattersJordan[17]. In Section 5.2 we prove some structural results for these rings and in particular an analogue of the decomposition R snT for R a UFD. Finally section 5.3 briefly sketches two other variations on the theme of unique factorisation due primarily to Cohn[ 20], and Beauregard [4], and shows that in general these theories are distinct.
