Let $R \hookrightarrow S$ be a finite extension of noetherian rings. Let $I \subseteq S$ be an $R$-submodule of $S$. Are there any sufficient criteria on $I$ such that it is in fact an ideal of $S$? Maybe in terms of homology or the geometry of the induced map $\textrm{Spec}(S) \to \textrm{Spec}(R)$?
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