Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-25T20:40:24.718Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-commutative Noetherian Unique Factorization Domains often have stable range one

Published online by Cambridge University Press:  28 June 2011

Martin Gilchrist
Martin Gilchrist
Affiliation:
Oxford University Press, Walton Street, Oxford OX2 6DP
Get access Rights & Permissions [Opens in a new window]

Extract

Let R be a Noetherian ring. If R is commutative then (a) every non-unit is contained in a height-1 prime ideal - this is the Krull principal ideal theorem - but (b) R can have an arbitrarily large stable range. The main aim of this paper is to show that if certain non-commutative rings satisfy condition (a) then they have stable range one. We shall prove the

Theorem. Let R be a Noetherian domain which is not commutative. Suppose that every non-unit of R lies in at least one height-1 prime ideal and that every height-1 prime is completely prime. Then the stable range of R is one.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 2 , September 1989 , pp. 229 - 235
Copyright
Copyright © Cambridge Philosophical Society 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Chatters, A. W.. Non-commutative Unique Factorization Domains. Math. Proc. Cambridge Philos. Soc. 95 (1984), 49–54.CrossRefGoogle Scholar
[2] Chatters, A. W. and Hajarnavis, C.. Rings urith chain conditions. Research Notes in Math. no. 44 (Pitman, 1982).Google Scholar
[3] Gilchrist, M. P. and Smith, M. K.. Non-commutative UFDs are often PIDs. Math. Proc. Cambridge Philos. Soc. 95 (1984), 417–419.Google Scholar
[4] Heath-Brown, R.. The divisor function at consecutive integers. Mathematika 31 (1984), 141–149.CrossRefGoogle Scholar
[5] Jategaonkar, A. V.. Injective modules and classical localization in Noetherian rings. Bull. Amer. Math. Soc. 79 (1973), 152–157.CrossRefGoogle Scholar
[6] Jategaonkar, A. V.. Localisation in Noetherian Rings. London Mathematical Society Lecture Note Series no. 97 (Cambridge University Press, 1986).CrossRefGoogle Scholar
[7] McConnell, J. C. and Robson, J. C.. Non-commutative Noetherian Rings (Wiley, 1987).Google Scholar
[8] Robson, J. C.. Idealisers and hereditary Noetherian prime rings. J. Algebra 22 (1972), 45–81.CrossRefGoogle Scholar
[9] Stafford, J. T.. The Goldie rank of a module. In Noetherian Rings and their Applications, Amer. Math. Soc. Math. Surveys no. 24 (1988), pp. 1–20.Google Scholar
[10] Warfield, R.. Cancellation of modules and groups and stable range of endomorphism rings. Pacific J. Math. 91 (1980), 457–485.Google Scholar