Kronecker function rings and flat $D[X]$-modules
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- by J. T. Arnold and J. W. Brewer
- Proc. Amer. Math. Soc. 27 (1971), 483-485
- DOI: https://doi.org/10.1090/S0002-9939-1971-0289489-0
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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.References
- Robert W. Gilmer, Multiplicative ideal theory, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR 0229624
- Robert Gilmer, An embedding theorem for $\textrm {HCF}$-rings, Proc. Cambridge Philos. Soc. 68 (1970), 583–587. MR 263792, DOI 10.1017/s0305004100076568
- Fred Richman, Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965), 794–799. MR 181653, DOI 10.1090/S0002-9939-1965-0181653-1
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 27 (1971), 483-485
- MSC: Primary 13.50
- DOI: https://doi.org/10.1090/S0002-9939-1971-0289489-0
- MathSciNet review: 0289489