Suppose $R=S/I$ is the homomorphic image of a local ring $(S,m)$. I know the completion of $R$ as an $S$-module is just $R \otimes \hat S$, where $\hat S$ is the $m$-adic completion of $S$. Is this also the completion of $R$ as a ring? (I think this comes down to showing that the inverse limit of $S/(m^i +I)$ is the same as the inverse limit of $S/m^i$ tensored with $S/I$.)
Completion of the homomorphic image of a local ring.
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ring-theory
commutative-algebra