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

1. $M$ is torsion free,

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

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

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

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

Proof. We have seen the equivalence of (1) and (2) in Lemma 15.22.7. We have seen the equivalence of (4) and (5) in Algebra, Lemma 10.157.2. The equivalence between (3) and (5) is immediate from the definition. A localization of a torsion free module is torsion free (Lemma 15.22.3), hence it is clear that a $M$ has no associated primes different from $(0)$. Thus (1) implies (5). Conversely, assume (5). If $M$ has torsion, then there exists an embedding $R/I \subset M$ for some nonzero ideal $I$ of $R$. Hence $M$ has an associated prime different from $(0)$ (see Algebra, Lemmas 10.63.3 and 10.63.7). This is an embedded associated prime which contradicts the assumption. $\square$

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