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

Authors: William Heinzer and Jack Ohm
Journal: Proc. Amer. Math. Soc. 35 (1972), 1-8
MSC: Primary 13C05
MathSciNet review: 0306177
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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 ``$ R[X]/I$ is R-flat implies I is finitely generated'' 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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Keywords: Polynomial ring, flat module, finitely generated ideal, prime ideal
Article copyright: © Copyright 1972 American Mathematical Society

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