It is well-known that for finitely generated modules over a Dedekind domain, flatness and torsion-free are equivalent. Is this true for general Noetherian rings? If not, where is the dimension one property used?
2026-03-25 14:36:49.1774449409
Dedekind domain necessary for equivalence of flatness and torsion-free
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In fact, the property has nothing to do with dimension (it is actually a homological property), but with special properties of ideals. If $R$ is a domain, then your property is equivalent with $R$ Prüfer domain, and noetherian Prüfer domains are Dedekind.