Lemma 10.96.6. Let $R$ be a ring, let $I \subset R$ be an ideal, and let $R^\wedge = \mathop{\mathrm{lim}}\nolimits R/I^ n$.

any element of $R^\wedge $ which maps to a unit of $R/I$ is a unit,

any element of $1 + I$ maps to an invertible element of $R^\wedge $,

any element of $1 + IR^\wedge $ is invertible in $R^\wedge $, and

the ideals $IR^\wedge $ and $\mathop{\mathrm{Ker}}(R^\wedge \to R/I)$ are contained in the Jacobson radical of $R^\wedge $.

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