Wilson, Dean David (1989) Non-Noetherian unique factorisation rings. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Noncommutative rings, Rings (Algebra), Factorization (Mathematics), Commutative rings
Official Date:	September 1989
Dates:	Date Event September 1989 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Hajarnavis, C. R
Sponsors:	Science and Engineering Research Council (Great Britain)
Extent:	[6], 95 leaves
Language:	eng

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