Trees and valuation rings

Abstract:

A subring $B$ of a division algebra $D$ is called a valuation ring of $D$ if $x\in B$ or $x^{-1}\in B$ holds for all nonzero $x$ in $D$. The set $\mathcal {B}$ of all valuation rings of $D$ is a partially ordered set with respect to inclusion, having $D$ as its maximal element. As a graph $\mathcal {B}$ is a rooted tree (called the valuation tree of $D$), and in contrast to the commutative case, $\mathcal {B}$ may have finitely many but more than one vertices. This paper is mainly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra $D$, and one main result here is a positive answer to this question where $D$ can be chosen as a quaternion division algebra over a commutative field.