Quotient of complete topological ring Let $G$ be a complete topological group (meaning that every Cauchy net has a unique limit), and $H\unlhd G$ a closed normal subgroup. If $G$ is first countable (equivalently, metrizable), then $G/H$ is complete, but there are examples to show that $G/H$ can fail to be complete if $G$ is not first countable.
I'm wondering if there are similar examples for topological rings and closed ideals. A commutative topological ring $R$ is said to be linearly topologized if there is a neighborhood basis of 0 consisting of ideals.

Does there exist a complete linearly topologized commutative ring $R$, and a closed ideal $I\subset R$, such that $R/I$ is not complete?

Such an $R$ must necessarily not be first countable.
I asked this question on MSE last week, but there were no answers.
 A: This is not an answer but rather a sketch how I believe a counterexample can be constructed (I do not like to pose such an incomplete answer but I cannot do better at the moment).

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*I am quite sure that, for a suitable directed set $I$, there is a projective $I$-spectrum $\mathscr X=(X_\alpha,\varrho_\beta^\alpha)$ of abelian groups such that the induced morphisms on the limit $\varrho_\infty^\alpha: \lim\limits_{\leftarrow} \mathscr X \to X_\alpha$ are surjective but the derived projective limit functor ${\lim\limits_{\leftarrow}}^{(1)} \mathscr X$ does not vanish.

Edit. An example is in these notes https://home.mathematik.uni-freiburg.de/ziegler/preprints/invers-limit.pdf of M. Ziegler who attributes it to S. Todorcevic (unfortunately, without a reference).

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*The resolution $$0\to\lim\limits_{\leftarrow} \mathscr X \to \prod_{\alpha\in I} X_\alpha \stackrel{d_0}{\longrightarrow} \prod_{\alpha\le\beta} X_\alpha \stackrel{d_1}{\longrightarrow} \cdots$$ to compute the derived functors will thus fail to be exact at the spot $\prod_{\alpha\le\beta} X_\alpha$.


*Endow all $X_\alpha$ with the discrete topology and the product $\prod_{\alpha\in I}X_\alpha$ with the product topology which has a basis of the zero-neighbourhood filter consisting of subgroups.


*The surjectivity of $\varrho_\infty^\alpha$ should imply that $d_0$ is open on its image which is dense in the kernel of $d_1$. Hence, the image of $d_0$ is an incomplete quotient of the product.


*To get an example of linearly topologized rings just take the zero product on all groups $X_\alpha$.
In particular, the last point might be disappointing (one can probably hide this triviality a bit by adjoining units). If someone wants to study the details of this sketch, I recommend

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*C.U. Jensen: Les foncteurs dérivés de  $\lim\limits_{\leftarrow}$ et leurs applications en théorie des
modules. Springer Lecture Notes in Mathematics 254.


*B. Mitchell: Rings with several objects. Adv. Math., 8:1–161, 1972.


*J.Wengenroth: Derived Functors in Functional Analysis, Springer Lecture Notes in Mathematics 1810.
The lecure notes of Jensen contain results about Artinian modules which might be interesting to exclude the trivial multiplication.
