$\def\fra{\mathfrak{a}}$Here it is proven that for $A$ a Noetherian ring, $\fra\subset A$ an ideal and $M$ a finitely-generated $A$-module and if we take $\fra$-adic completions, then $\hat{\mathfrak{a}}\hat{M}=\widehat{\mathfrak{a}M}$.

In general, it is always true that $\hat{\mathfrak{a}}\hat{M}\subset\widehat{\mathfrak{a}M}$. I was wondering:

What's an example of a module $M$ for which $\hat{\mathfrak{a}}\hat{M}\not=\widehat{\mathfrak{a}M}$?

The instances I know in the literature that try to exemplify related behaviors in the non-Noetherian case (here and Bourbaki, Commutative Algebra, Ch. III, Exercise 2.12, p. 235) are all done for $M=A$; and the thing is that on this case we always have $\hat{\mathfrak{a}}\hat{M}=\widehat{\mathfrak{a}M}$. So what does a counterexample look like?



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