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Let $R$ be a non-commutative ring equipped with an $I$-adic filtration, where $I$ is a prime ideal. Consider the set $S=1+I$ and assume that $S$ is a left and right Ore set. Therefore one may form the ring $S^{-1}R$, which comes equipped with an induced $I$-adic filtration.

Consider the $I$-adic completion of $S^{-1}R$, denoted $\widehat{S^{-1}R}$. Is it true that $\widehat{S^{-1}R}=\widehat{R}$. My intuition comes from the fact that every element of $S$ is now a unit in the completion.

Looking for hints and references.

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