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

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.