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

 

Characterizing nearly simple chain domains


Authors: H. H. Brungs and J. Gräter
Journal: Proc. Amer. Math. Soc. 131 (2003), 1347-1355
MSC (2000): Primary 16L30, 16N60, 16D25; Secondary 06D99.
Published electronically: September 19, 2002
MathSciNet review: 1949863
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Abstract: G. Puninski, using model theoretical methods, showed that if a chain domain $R$ is nearly simple, then $Ra+bR = J(R)$ for any nonzero elements $a,b$ in $J(R)$, the Jacobson radical of $R$. Here, an algebraic proof is given for this result, exceptional chain domains are characterized, and it is shown that $V_0(R)$, the lattice generated by all proper nonzero left and right ideals, is a direct product of two linearly ordered sets if $R$ is nearly simple. In a certain sense this property characterizes nearly simple chain domains among all integral domains.


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Additional Information

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

J. Gräter
Affiliation: Institut für Mathematik, Universität Potsdam, Postfach 601553, 14469 Potsdam, Germany
Email: graeter@rz.uni-potsdam.de

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06645-5
PII: S 0002-9939(02)06645-5
Received by editor(s): August 6, 2001
Received by editor(s) in revised form: December 17, 2001
Published electronically: September 19, 2002
Additional Notes: The first author was supported in part by NSERC
Communicated by: Martin Lorenz
Article copyright: © Copyright 2002 American Mathematical Society