## The finiteness of $I$ when $R[X]/I$ is $R$-flat. II

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- by William Heinzer and Jack Ohm
- Proc. Amer. Math. Soc.
**35**(1972), 1-8 - DOI: https://doi.org/10.1090/S0002-9939-1972-0306177-3
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## Abstract:

This paper supplements work of Ohm-Rush. A question which was raised by them is whether $R[X]/I$ is a flat*R*-module implies

*I*is locally finitely generated at primes of $R[X]$. Here

*R*is a commutative ring with identity,

*X*is an indeterminate, and

*I*is an ideal of $R[X]$. It is shown that this is indeed the case, and it then follows easily that

*I*is even locally principal at primes of $R[X]$. Ohm-Rush have also observed that a ring

*R*with the property â$R[X]/I$ is

*R*-flat implies

*I*is finitely generatedâ is necessarily an $A(0)$ ring, i.e. a ring such that finitely generated flat modules are projective; and they have asked whether conversely any $A(0)$ ring has this property. An example is given to show that this conjecture needs some tightening. Finally, a theorem of Ohm-Rush is applied to prove that any

*R*with only finitely many minimal primes has the property that $R[X]/I$ is

*R*-flat implies

*I*is finitely generated.

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## Bibliographic Information

- © Copyright 1972 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**35**(1972), 1-8 - MSC: Primary 13C05
- DOI: https://doi.org/10.1090/S0002-9939-1972-0306177-3
- MathSciNet review: 0306177