Let be a finitely generated module over a Dedekind domain. Prove that is flat is torsion free.
: let and consider the R-homomorphism defined by since R is an integral domain, is injective.
thus the map must be injective too because M is assumed to be flat. but so the map is basically
the map defined by: now we have thus since is injective. therefore M is torsion free.
: since R is a Dedekind domain and M is finitely generated and torsion free, M is isomorphic to a direct sum of ideals of R. so M is projective, because every
ideal of a Dedekind domain is projective. finally this trivial fact that every projective module is flat completes the proof. Q.E.D.
Remark 1: for we only needed R to be a domain.
Remark 2: the equivalence of "torsion free" and "flat" holds for any module (not necessarily finitely generated) over a Dedekind domain.