Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-09T07:55:04.219Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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].

Information

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

References

1.Chatters, A. W., Non-commutative unique factorisation domains, Math. Proc. Cambridge Philos. Soc. 95 (1984), 49–54.CrossRefGoogle Scholar
2.Chatters, A. W. and Hajarnavis, C. R., Rings with chain conditions (Pitman, 1980).Google Scholar
3.Chatters, A. W. and Jordan, D. A., Non-commutative unique factorisation rings, J. London Math. Soc. (2) 33 (1986), 22–32.CrossRefGoogle Scholar
4.Cohn, P. M., Free rings and their relations, London Math. Soc. Monograph No. 19, 2nd edition (Academic Press, 1985).Google Scholar
5.Cozzens, J. H. and Sandomierski, F. L., Reflexive primes, localization and primary decomposition in maximal orders, Proc. Anter. Math. Soc. 58 (1976), 44–50.CrossRefGoogle Scholar
6.Goldie, A. W. and Michler, G. O., Ore extensions and polycyclic group rings, J. London Math. Soc. (2) 9 (1974), 337–345.CrossRefGoogle Scholar
7.Jordan, D. A., Noetherian Ore extensions and Jacobson rings, J. London Math. Soc. (2) 10 (1975), 281–291.CrossRefGoogle Scholar
8.Jordan, D. A., Primitive Ore extensions, Glasgow Math. J. 18 (1977), 93–97.CrossRefGoogle Scholar
9.Jordan, D. A., Normal elements and completions of non-commutative Noetherian rings, Bull. London Math. Soc. 19 (1987), 417–424.CrossRefGoogle Scholar
10.Kaplansky, I., Commutative rings, revised edition (University of Chicago Press, 1974).Google Scholar
11.Maury, G. and Raynaud, J., Ordres maximaux au sens de K. Asano, Lecture Notes in Mathematics 808 (Springer, 1980).CrossRefGoogle Scholar
12.Samuel, P., Anneaux factoriels (Sociedade de Matemática de Sāo Paulo, 1963).Google Scholar