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

DOI:
https://doi.org/10.1090/S0002-9939-00-05440-X

Published electronically:
May 18, 2000

MathSciNet review:
1690977

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a simple artinian ring. A valuation ring of is a Bézout order of so that is simple artinian, a Goldie prime is a prime ideal of so that is Goldie, and a prime segment of is a pair of neighbouring Goldie primes of A prime segment is archimedean if is equal to it is simple if and it is exceptional if In this last case, is a prime ideal of so that 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 so that is not divisorial for

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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:
https://doi.org/10.1090/S0002-9939-00-05440-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