A NOTE ON VALUED DIVISION-ALGEBRAS

Let D be a finite-dimensional division algebra with center F and let B be a valuation ring of D, i.e., xεB or x-1εB holds for all nonzero x in D. Then B is called an extension of V=F∈B. Let ß be the set of all extensions of V to D and let P(B) be the group of all bijections of ß which are induced by the inner automorphisms of D. In this note we investigate P(B). It is shown that P(B) is solvable and an upper bound for {norm of matrix}P(B){norm of matrix} is given. Finally, Theorem 4.5 gives a necessary and sufficient condition for B such that {norm of matrix}P(B){norm of matrix} is equal to this bound. © 1992 Academic Press, Inc.