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I would like to know how are the valuation rings of complete non-archimedean fields which are not local. Take, for example, $\mathbb{C}_p$, and its valuation ring, $\mathcal{O}_{\mathbb{C}_p}=\{x\in\mathbb{C}_p|v(x)\geq0\}$.

Since it is valuation ring, then it is local and integrally closed. But is it Noetherian? And does it have Krull dimension 1? And more generally, for any field $K$ like the ones given?

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up vote 2 down vote accepted

A valuation ring is always local, and a valuation ring is Noetherian if and only if it is a discrete valuation ring.

EDIT: I realize now that you aren't asking if there are non-local valuation rings, you're asking about the valuation rings of non-Archimedean fields which themselves aren't local, i.e., locally compact. They are always $1$-dimensional, but they are Noetherian if and only if discretely valued.

A valuation ring has rank $1$ in the sense that its value group is isomorphic to a subgroup of $\mathbf{R}$ if and only if it is $1$-dimensional. This is Theorem 10.7 in Matsumura's Commutative ring theory. So the valuation ring of $\mathbf{C}_p$, while definitely not Noetherian (because $\mathfrak{m}=\mathfrak{m}^2$), is $1$-dimensional, and more generally, the valuation ring of any non-Archimedean, non-trivially valued field is $1$-dimensional.

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