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 R[X]/I is a flat R-module implies I is locally finitely generated at primes of R [ X ] R[X] . Here R is a commutative ring with identity, X is an indeterminate, and I is an ideal of R [ X ] 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 ] R[X] . Ohm-Rush have also observed that a ring R with the property “ R [ X ] / I R[X]/I is R-flat implies I is finitely generated” is necessarily an A ( 0 ) A(0) ring, i.e. a ring such that finitely generated flat modules are projective; and they have asked whether conversely any A ( 0 ) 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 R[X]/I is R-flat implies I is finitely generated.