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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
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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

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