Let consider $A=k[x_{1},x_{2}...]$, the polynomial ring with countably many indeterminates.
Then we can consider the completion $\hat{A}=\varprojlim_{r,l}k[x_{1},x_{2},..]/(x_{1}^{r},..x_{l}^{r},x_{l+1},x_{l+2}..)$. In particular, it is shown in
Is $R=k[x_1,\ldots]\to k[[x_1,\ldots]]$ a flat morphism? What about $R\to\hat{R}$?
that $\hat{A}$ is faithfully flat.
Let $\mathfrak{m}:=(x_{1},x_{2},...)$ and $\hat{\mathfrak{m}}:=ker(\hat{A}\rightarrow A/\mathfrak{m})$.
What can we say about $\hat{\mathfrak{m}}/\mathfrak{m}\hat{A}$, for example is it artinian?