Lemma 15.23.11. Let $R$ be a Noetherian ring. Let $M$, $N$ be finite $R$-modules.

If $N$ has property $(S_1)$, then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ has property $(S_1)$.

If $N$ has property $(S_2)$, then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ has property $(S_2)$.

If $R$ is a domain, $N$ is torsion free and $(S_2)$, then $\mathop{\mathrm{Hom}}\nolimits _ R(M, N)$ is torsion free and has property $(S_2)$.

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