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.


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 ...


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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