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Unfortunately, the Krull dimension often behaves in strange ways for such non-noetherian rings. My impression is that the results of Lang--Ludwig should adapt to the present situation, and show that this ring has infinite Krull dimension.

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This is essentially answered in one of the answers to Is every compact topological ring a profinite ring?. If a compact ring $R$ either admits no element $r\neq 0$ with $rR=0$ or the left-right dual condition then it is profinite. This is the condition that the multiplication map induces and embedding of $R$ into the endomorphisms of the Pontryagin dual of ...

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If I am correct, your Noetherian local domain $(R,m(R))$ admits a place $D$ (maximal local domain) such that $R\subset D\subset K$ and $m(D)\cap R=m(R)$ (for a local ring $L$, we note $m(L)$ its maximal ideal). The fraction field $K$ is common to $R$ and $D$. As $D$ is a place there is a valuation $$v\ :\ K\to \Gamma_{\infty}$$ such that  D=\{x\in K\,|\...

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