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Unique factorisation of normal elements in non-commutative rings

Published online by Cambridge University Press:  18 May 2009

D. A. Jordan
D. A. Jordan
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH
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In the literature there are several generalisations to non-commutative rings of the notion of a unique factorisation domain from commutative algebra. This paper follows in the spirit of [1, 3] and is set in the context of Noetherian rings. In [3], A. W. Chatters and the author denned a Noetherian UFR (unique factorisation ring) to be a prime Noetherian ring R in which every non-zero prime ideal contains a prime ideal generated by a non-zero normal element p, that is, by an element p such that pR = Rp. The class of Noetherian UFRs includes the Noetherian UFDs studied by Chatters in [1], while a commutative Noetherian ring is a UFR if and only if it is a UFD in the usual sense. For a Noetherian UFR R, the following are simple consequences of the definition:

(i) every non-zero ideal of R contains a non-zero normal element;

(ii) the set N(R) of non-zero normal elements of R is a unique factorisation monoid in the sense of [4, Chapter 3].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 1 , January 1989 , pp. 103 - 113
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

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