The Stacks project

Tag 00MC doesn't need $(R,\mathfrak m)$ to be local. The result holds as long as $I$ is contained in the Jacobson radical of $R$. Perhaps it could be stated this way since the proof does not require any serious modifications?

Much better. Thanks! Changes are here.

Alternative suggested proof: By Lemma 10.97.2 it is flat. Thus, if $M$ is some non-zero $R$-module, to see that $M\otimes_R\hat{R}\neq 0$ it suffices to see that $N\otimes_R\hat{R}\neq 0$, for finitely generated submodules $N\subset M$. By Lemma 10.97.1, $N\otimes_R\hat{R}\cong\hat{N}$. Since $\hat{N}=\varprojlim_n N/I^nN$, the map $N\to \hat{N}$ is injective by Krull's intersection theorem (Lemma 10.51.4), so $N\neq 0$ implies $\hat{N}\neq 0$.