Definition 10.96.2. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$-module. We say $M$ is $I$-adically complete if the map

$M \longrightarrow M^\wedge = \mathop{\mathrm{lim}}\nolimits _ n M/I^ nM$

is an isomorphism1. We say $R$ is $I$-adically complete if $R$ is $I$-adically complete as an $R$-module.

[1] This includes the condition that $\bigcap I^ nM = (0)$.

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