Valuation rings in finite-dimensional division algebras

Abstract

A subring B′ of a division algebra D is called a valuation ring of D if x or x −1 is contained in B′ for every x in D, x ≠ 0. Such a ring is called an extension of the valuation ring B of K, the centre of D, if B′ ∩ K = B. Let D be a division algebra finite-dimensional over its centre K, [ D : K] = n 2, B a valuation ring of K and B = { B i| iε I} the set of all extension of B in D. Theorem 1. B possesses at most n extensions in D, i.e., | B |⩽ n. Theorem 2. Any two extensions of B in D are conjugate in D. Theorems 3 and 4. The set T of elements in D which are integral over B is a subring of D if and only if ¦ B ¦⩾ 1. In this case T = ∩ B i , B i , in B .