Abstract
A subring B of a division algebra D is called a valuation ring of D if x E B or x1 E B holds for all nonzero x in D. The set 13 of all valuation rings of D is a partially ordered set with respect to inclusion, having D as its maximal element. As a graph 13 is a rooted tree (called the valuation tree of D), and in contrast to the commutative case, 13 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 fielcl.
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