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Assume $S= R[T]/(f)= R[w]$ is a flat non-divisible $R$-module, where $R$ is a noetherian UFD, $T$ is an indeterminate over $R$, and $f\in R[T]$ is a non-monic polynomial of positive degree.

Can we say if $S$ is $R$-projective or not?

I guess there is not enough information to answer this question. If so, is there an additional condition which guarantees the projectivity of $S$ as an $R$-module?

This question is somewhat relevant. I have now posted this question here also (willing to delete one of them).

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