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Unique factorisation rings

Published online by Cambridge University Press:  20 January 2009

A. W. Chatters ,
M. P. Gilchrist  and
D. Wilson
A. W. Chatters
Affiliation:
School of MathematicsUniversity of BristolUniversity WalkBristol BS8 1TN
M. P. Gilchrist
Affiliation:
Science and Medical DepartmentOxford University PressWalton StreetOxford OX2 6DP
D. Wilson
Affiliation:
Mathematical InstituteUniversity of WarwickCoventry CV4 7AL
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Abstract

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Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 35 , Issue 2 , June 1992 , pp. 255 - 269
Copyright
Copyright © Edinburgh Mathematical Society 1992

References

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