Extensions of valuation rings in central simple algebras

Abstract:

Certain subrings $R$ of simple algebras $Q$, finite dimensional over their center $K$, are studied. These rings are called $Q$-valuation rings since they share many properties with commutative valuation rings. Let $V$ be a valuation ring of $K$, the center of $Q$, and let $\mathcal {R}$ be the set of $Q$-valuation rings $R$ in $Q$ with $R \cap K = V$, then $\left | \mathcal {R} \right | \geq 1$. 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 $\mathcal {R}$ is obtained. Theorem. The elements in $\mathcal {R}$ are conjugate if $V$ has finite rank. Theorem. The elements in $\mathcal {R}$ are total valuation rings if $\mathcal {R}$ contains one total valuation ring. In this case $Q$ is a division ring. Theorem. $\mathcal {R}$ if $\mathcal {R}$ contains an invariant total valuation ring.