The polynomial extension $R \rightarrow R[X]$ ($X$ an indeterminate) has many nice properties beyond faithful flatness. The one I'm most interested in at the moment is the following. Say that a flat $R$-algebra $S$ is *intersection flat* (terminology due to Hochster?) or a *content module* (terminology due to Ohm and Rush), or an algebra that *has content* (terminology of Eakin and Silver) if for any collection $\{I_\alpha\}_{\alpha \in \Lambda}$ of ideals $I_\alpha$ of $R$, one has $\bigcap_{\alpha \in \Lambda} (I_\alpha S) = (\bigcap_{\alpha \in \Lambda} I_\alpha)S$.

For this to make sense, of course, you don't need $S$ to be an $R$-algebra; it can be an $R$-module.

Does this property have any meaning in algebro-geometric terms? I ask because I want examples of it beyond the case where $S$ is projective as an $R$-module, or $S=R[[X]]$, or various kinds of localizations of same. Actually I want examples that satisfy some additional properties (e.g. I also require that primes extend to primes), but I'm probably getting ahead of myself.

By the way, I'm looking for pretty general constructions that will work over a large class of base rings. Any examples where $R$ has to be a field or a PID are useless to me, for instance.

Thanks!

EDIT: Maybe I should exhibit some additional cases where this condition holds. It holds, for instance, whenever $R$ is complete Noetherian and $R \rightarrow S$ is a flat local homomorphism (by Chevalley's theorem). Also, it always holds for the extension $R \rightarrow R[[x]]$ (though there, primes do not typically exend to primes), and when $R=k[X_1, \dotsc, X_n]$, with $k$ a perfect field of prime characteristic, then the Frobenius endomorphism is intersection flat (though there again, primes almost never extend to primes).