Lemma 10.69.9. Let $R$ be a ring. Let $J = (f_1, \ldots , f_ r)$ be an ideal of $R$. Let $M$ be an $R$-module. Set $\overline{R} = R/\bigcap _{n \geq 0} J^ n$, $\overline{M} = M/\bigcap _{n \geq 0} J^ nM$, and denote $\overline{f}_ i$ the image of $f_ i$ in $\overline{R}$. Then $f_1, \ldots , f_ r$ is $M$-quasi-regular if and only if $\overline{f}_1, \ldots , \overline{f}_ r$ is $\overline{M}$-quasi-regular.

Proof. This is true because $J^ nM/J^{n + 1}M \cong \overline{J}^ n\overline{M}/\overline{J}^{n + 1}\overline{M}$. $\square$

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