Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quasi-valuations and algebras over valuation domains   Suppose $F$ is a field with valuation $v$ and valuation domain $O_{v}$, and
$R$ is an $O_{v}-$algebra. We prove that $R$ satisfies SGB (strong going
between) over $O_{v}$. We give a necessary and sufficient condition for $R$ to
satisfy LO (lying over) over $O_{v}$. Using the filter \qv constructed in
[Sa1], we show that if $R$ is torsion-free over $O_{v}$ then $R$ satisfies GD
(going down) over $O_{v}$. In particular, if $R$ is torsion-free and
$(R^{\times} \cap O_{v}) \subseteq O_{v}^{\times}$, then for any chain in
$\text{Spec}(O_v)$ there exists a chain in $\text{Spec}(R)$ covering it.
  Assuming $R$ is torsion-free over $O_{v}$ and $[R \otimes_{O_{v}}F:F]<
\infty$, we prove that $R$ satisfies INC (incomparabilty) over $O_{v}$.
Assuming in addition that $(R^{\times} \cap O_{v}) \subseteq O_{v}^{\times}$,
we deduce that $R$ and $O_{v}$ have the same Krull dimension and a bound on the
size of the prime spectrum of $R$ is given.
  Under certain assumptions on $R$ and a \qv defined on it, we prove that the
\qv ring satisfies GU (going up) over $O_{v}$. Combining these five properties
together, we deduce that any maximal chain of prime ideals of the \qv ring is
lying over $\text{Spec}(O_{v})$, in a one-to-one correspondence.

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The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys