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

 

Kronecker function rings and flat $ D[X]$-modules


Authors: J. T. Arnold and J. W. Brewer
Journal: Proc. Amer. Math. Soc. 27 (1971), 483-485
MSC: Primary 13.50
MathSciNet review: 0289489
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Abstract: Let $ D$ be an integral domain with identity. Gilmer has recently shown that in order that a $ v$-domain $ D$ be a Prüfer $ v$-multiplication ring, it is necessary and sufficient that $ {D^v}$ be a quotient ring of $ D[X]$, where $ {D^v}$ is the Kronecker function ring of $ D$ with respect to the $ v$-operation. In this paper the authors prove that in the above theorem it is possible to replace ``a quotient ring of $ D[X]$'' with ``a flat $ D[X]$-module.'' Moreover, it is shown that $ {D^v}$ is the only Kronecker function ring of $ D[X]$ which can ever be a flat $ D[X]$-module.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0289489-0
PII: S 0002-9939(1971)0289489-0
Keywords: Kronecker function ring, flat module, essential valuation ring, Prüfer $ v$-multiplication ring
Article copyright: © Copyright 1971 American Mathematical Society