Lemma 15.22.8. Let $R$ be a Noetherian domain. Let $M$ be a nonzero finite $R$-module. The following are equivalent

$M$ is torsion free,

$M$ is a submodule of a finite free module,

$(0)$ is the only associated prime of $M$,

$(0)$ is in the support of $M$ and $M$ has property $(S_1)$, and

$(0)$ is in the support of $M$ and $M$ has no embedded associated prime.

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