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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A classification of prime segments in simple artinian rings


Authors: H. H. Brungs, H. Marubayashi and E. Osmanagic
Journal: Proc. Amer. Math. Soc. 128 (2000), 3167-3175
MSC (1991): Primary 16W60; Secondary 16L30
Published electronically: May 18, 2000
MathSciNet review: 1690977
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Abstract: Let $A$ be a simple artinian ring. A valuation ring of $A$ is a Bézout order $R$ of $A$ so that $R/J(R)$ is simple artinian, a Goldie prime is a prime ideal $P$ of $R$ so that $R/P$ is Goldie, and a prime segment of $A$ is a pair of neighbouring Goldie primes of $R.$A prime segment $P_{1}\supset P_{2}$ is archimedean if $K(P_{1})=\{a\in P_{1}\vert P_{1} aP_{1}\subset P_{1}\}$ is equal to $P_{1},$ it is simple if $K(P_{1})=P_{2}$and it is exceptional if $P_{1}\supset K(P_{1})\supset P_{2}.$ In this last case, $K(P_{1})$ is a prime ideal of $R$ so that $R/K(P_{1})$ is not Goldie. Using the group of divisorial ideals, these results are applied to classify rank one valuation rings according to the structure of their ideal lattices. The exceptional case splits further into infinitely many cases depending on the minimal $n$ so that $K(P_{1})^{n}$ is not divisorial for $n\ge 2.$


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Additional Information

H. H. Brungs
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: hbrungs@vega.math.ualberta.ca

H. Marubayashi
Affiliation: Department of Mathematics, Naruto University of Education, Naruto, Japan
Email: marubaya@naruto-u.ac.jp

E. Osmanagic
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: eosman@vega.math.ualberta.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05440-X
PII: S 0002-9939(00)05440-X
Keywords: Dubrovin valuation ring, local Bézout order, total valuation ring, Goldie prime, localizable prime, divisor group
Received by editor(s): December 29, 1997
Received by editor(s) in revised form: January 5, 1999
Published electronically: May 18, 2000
Additional Notes: The first author is supported in part by NSERC
Communicated by: Ken Goodearl
Article copyright: © Copyright 2000 American Mathematical Society