Let \({\mathcal D}\) be a finite-dimensional central division algebra over a field F. Suppose that v: \({\mathcal D}\to \Gamma\) is a Schilling valuation on \({\mathcal D}\) and \(\tau: {\mathcal D}\to {\mathcal D}\) is an involution. One says that v is a c-valuation (with respect to \(\tau\)) if for all \(a,a_ i,b\in {\mathcal D}^*\) (\({\mathcal D}^*\) is the group of nonzero elements of \({\mathcal D})\) \[ v(a^{\tau})=v(a),\quad v(\sum a_ ia^{\tau}_ i)=\min_ i\{v(aa^{\tau}_ i)\},\quad baa^{\tau}\equiv aa^{\tau}b\quad (mod(1+M)), \] where M is the ideal of the valuation v.
The main result of the paper under review is the following Theorem 1: Let \({\mathcal D}\) be any finite-dimensional c-valued division ring with \({\mathcal D}\neq F\) and suppose that
(H) given any \(a\in M\) there is a \(b\in F(a)\) such that \(1+a=b^ 2.\)
Then \({\mathcal D}\) decomposes into a tensor product of \(\tau\)-closed quaternion algebras over F. Observe that if F is Henselian and 2 doesn’t belong to M then any algebra \({\mathcal D}\) satisfies (H).
In the general case the authors prove Theorem 2. Every finite-dimensional c-valued ring has an immediate extension \({\mathcal D}_{\phi}\) which decomposes into a tensor product of \(\tau\)-closed quaternion division subrings. The second part of the paper is devoted to the consideration of the problem of constructing c-valued division algebras. Finally the authors show that not every c-valued division algebra is a tensor product of quaternion algebras.