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

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

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

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

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

Proof. Let $x \in R^\wedge$ map to a unit $x_1$ in $R/I$. Then $x$ maps to a unit $x_ n$ in $R/I^ n$ for every $n$ by Lemma 10.32.4. Hence $y = (x_ n^{-1}) \in \mathop{\mathrm{lim}}\nolimits R/I^ n = R^\wedge$ is an inverse to $x$. Parts (2) and (3) follow immediately from (1). Part (4) follows from (1) and Lemma 10.19.1. $\square$

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