Certain subrings
R
R
of simple algebras
Q
Q
, finite dimensional over their center
K
K
, are studied. These rings are called
Q
Q
-valuation rings since they share many properties with commutative valuation rings. Let
V
V
be a valuation ring of
K
K
, the center of
Q
Q
, and let
R
\mathcal {R}
be the set of
Q
Q
-valuation rings
R
R
in
Q
Q
with
R
∩
K
=
V
R \cap K = V
, then
|
R
|
≥
1
\left | \mathcal {R} \right | \geq 1
. This extension theorem, which does not hold if one considers only total valuation rings, was proved by N. I. Dubrovin. Here, first a somewhat different proof of this result is given and then information about the set
R
\mathcal {R}
is obtained. Theorem. The elements in
R
\mathcal {R}
are conjugate if
V
V
has finite rank. Theorem. The elements in
R
\mathcal {R}
are total valuation rings if
R
\mathcal {R}
contains one total valuation ring. In this case
Q
Q
is a division ring. Theorem.
R
\mathcal {R}
if
R
\mathcal {R}
contains an invariant total valuation ring.