There is a well-developed theory of unique factorization domains in commutative algebra. The generalization of this concept to non-commutative rings has also been extensively studied (e.g. in [19], [16], [1]). This thesis is concerned with classes of non-commutative rings which are generalizations of non-commutative Noetherian unique factorization rings. Noetherian UFR's are maximal orders and every reflexive ideal is invertible in these rings. Clearly maximal orders and reflexive ideals are important concepts and we examine them in this thesis. UFR's can also be considered as rings in which the divisor class group is trivial. This provides the motivation for us to study this group more generally. In Chapter 1, we give the basic material we shall need from the theory of noncommutative rings-and in Chapter 2, we present the known results about certain classes of rings which are crucial for this thesis. Chapters 3, 4 and 5 contain the original work of this thesis. In Chapter 3, we study the prime Noetherian maximal orders with enough invertible ideals. We show that in such rings every height 1 prime ideal is maximal reflexive and we prove results which generalize some of the results of Asano orders. In Chapter 4, we investigate divisor class groups. We also study the relations between these groups and the divisor -class group of the centre of the ring. In Chapter 5, we introduce the Generalized Dedekind prime rings (G-Dedekind prime rings), which are also a generalization of Noetherian UFR's, and we study this class of rings.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:502119 |
| Date | January 2008 |
| Creators | Akalan, Evrim |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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