### Some applications of the ultrafilter topology on spaces of valuation domains, Part I

In the present note we introduce the basic definitions and the main results concerning the spaces of valuation domains needed in the subsequent Part II.

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In the present note we introduce the basic definitions and the main results concerning the spaces of valuation domains needed in the subsequent Part II.

Let $K$ be a field and $A$ be a subring of $K$. In the present note, we present the main applications of the so called on the space $\mathrm{Zar}\left(K\right|A)$, introduced in the previous Part I. After recalling that $\mathrm{Zar}\left(K\right|A)$ is a spectral space, we give an explicit description of $\mathrm{Zar}\left(K\right|A)$ as the prime spectrum of a ring (even in the case when the quotient field of $A$ is a proper subfield of $K$). Moreover, we provide applications of the topological material previously introduced to the study of representations of integrally closed domains and...

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