Lemma 10.96.9. Let $R$ be a ring. Let $I$, $J$ be ideals of $R$. Assume there exist integers $c, d > 0$ such that $I^ c \subset J$ and $J^ d \subset I$. Then completion with respect to $I$ agrees with completion with respect to $J$ for any $R$-module. In particular an $R$-module $M$ is $I$-adically complete if and only if it is $J$-adically complete.

Proof. Consider the system of maps $M/I^ nM \to M/J^{\lfloor n/d \rfloor }M$ and the system of maps $M/J^ mM \to M/I^{\lfloor m/c \rfloor }M$ to get mutually inverse maps between the completions. $\square$

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