Lemma 10.68.8. Let $R$ be a ring. Let $0 \to M_1 \to M_2 \to M_3 \to 0$ be a short exact sequence of $R$-modules. Let $f_1, \ldots , f_ r \in R$. If $f_1, \ldots , f_ r$ is $M_1$-regular and $M_3$-regular, then $f_1, \ldots , f_ r$ is $M_2$-regular.

**Proof.**
By Lemma 10.4.1, if $f_1 : M_1 \to M_1$ and $f_1 : M_3 \to M_3$ are injective, then so is $f_1 : M_2 \to M_2$ and we obtain a short exact sequence

\[ 0 \to M_1/f_1M_1 \to M_2/f_1M_2 \to M_3/f_1M_3 \to 0 \]

The lemma follows from this and induction on $r$. Some details omitted. $\square$

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