The finiteness of when is -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 is a flat *R*-module implies *I* is locally finitely generated at primes of . Here *R* is a commutative ring with identity, *X* is an indeterminate, and *I* is an ideal of . It is shown that this is indeed the case, and it then follows easily that *I* is even locally principal at primes of .

Ohm-Rush have also observed that a ring *R* with the property `` is *R*-flat implies *I* is finitely generated'' is necessarily an ring, i.e. a ring such that finitely generated flat modules are projective; and they have asked whether conversely any 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 is *R*-flat implies *I* is finitely generated.

**[B]**N. Bourbaki,*Éléments de mathématique*. Fasc. XXVII.*Algèbre commutative*. Chaps. 1, 2, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR**36**#146.**[CP]**S. H. Cox Jr. and R. L. Pendleton,*Rings for which certain flat modules are projective*, Trans. Amer. Math. Soc.**150**(1970), 139–156. MR**0262296**, 10.1090/S0002-9947-1970-0262296-4**[G]**Robert W. Gilmer,*Multiplicative ideal theory*, Queen’s Papers in Pure and Applied Mathematics, No. 12, Queen’s University, Kingston, Ont., 1968. MR**0229624****[K]**Irving Kaplansky,*Commutative rings*, Allyn and Bacon, Inc., Boston, Mass., 1970. MR**0254021****[N]**Masayoshi Nagata,*Local rings*, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR**0155856****[OR]**Jack Ohm and David E. Rush,*The finiteness of 𝐼 when 𝑅[𝑋]/𝐼 is flat*, Bull. Amer. Math. Soc.**77**(1971), 793–796. MR**0279091**, 10.1090/S0002-9904-1971-12808-2**[V]**Wolmer V. Vasconcelos,*Simple flat extensions*, J. Algebra**16**(1970), 105–107. MR**0265342**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0306177-3

Keywords:
Polynomial ring,
flat module,
finitely generated ideal,
prime ideal

Article copyright:
© Copyright 1972
American Mathematical Society