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Title: Non-Noetherian unique factorisation rings
Author: Wilson, Dean David
ISNI:       0000 0001 3570 1200
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 1989
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The main aim of this thesis is to produce and then study two generalizations of the unique factorisation domain of commutative algebra. When this has been done before, [2] and [5], it has always been assumed that the rings are Noetherian. lt is our aim to show that this is not only unnatural but unnecessary. Chapter 1 contains some well known results about rings and in particular about rings satisfying a polynomial identity. In chapter 2 we define the unique factorisation ring (U.F.R.) and the unique factorisation domain (U.F.D.).show where these definitions come from and show what results can be obtained using only the definitions. In chapter 3 we show that all the previously known results for Noetherian U.F.D.s can be proved for a U.F.D. which merely satisfies the Goldie condition. In particular we prove that a Goldie U.F.D. is a maximal order and that a bounded Goldie U.F.D. is either commutative or a Noetherian principal ideal ring. In chapter 4 we look at U.F.R.s that satisfy a polynomial identity and show that these too are maximal orders. We also show that they are equal to the intersection of two rings .one of which is a Noetherian principal ideal ring and the other of which is a simple Artinian ring. In chapter 5 we look at the reflexive ideals of a U.F.R. which satisfies a polynomial identity and show that they are all principal. We also show that if T is a reflexive ideal of R then R/T has a quotient ring which is an Artinian principal ideal ring.
Supervisor: Not available Sponsor: Science and Engineering Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics