Let $R$ be a commutative ring, $T$ a variable, $f(T) \in R[T]$, $I=(f)$, $A = R[T]/I$ and $J$ the ideal of $R$ generated by the coefficients of $f$.
A special case of Nagata's flatness theorem (in his notations, $X=\{T\}$) says that $A$ is $R$-flat if and only if $J$ is a direct summand of $R$.
What happens if $I$ is non-principal? Is there a 'nice' result in this case?
For example, $R=\mathbb{C}[x^2,x^3]$, $A=\mathbb{C}[x]=R[T]/\tilde{I}$, $\tilde{I}$ is not principal, $\tilde{J}=R$, but $A$ is not $R$ flat; see the answer to this question ($\tilde{I}$ is generated by three polynomials).
Remark: I have read an article by one of Nagata's students concerning faithful flatness, in the same spirit of the above Nagata's flatness theorem, but unfortunately I am not able to find it now. Perhaps some ideas there are relevant for my question?
Thank you very much!
Edit: This is the article I referred to in the above remark.