Let $A$ be a commutative ring with unity and fix $f \in A$. Any $A$-module $M$ has its $f$-adic completion, the $\hat{A}$-module $\hat{M} = \underset{n}{\lim} M/f^nM$. There is a canonical map $\hat{A} \otimes_A M \to \hat{M}$, which is surjective if $M$ is finitely generated.
My question is about additional conditions which imply that this map is an isomorphism. Of course it is well-known that this is the case if $A$ is Noetherian. But I thought I remembered reading a paper where it was asserted that this bijectivity holds under some mild assumption like $f$-torsion-freeness (along with finite generation) of $M$. For instance, is $\hat{A} \otimes_A I \to \hat{I}$ an isomorphism if $I$ is a finitely generated ideal in a domain $A$?
I couldn't relocate this paper or reconstruct the argument myself, but maybe this rings a bell for someone on MO?
