Let K be a not necessarily commutative field and K x the multiplicative group of K. A subring B of K is called a vahtation rhzg if every element of K is either in B or is the inverse of an element of B.
A valuation ring B induces a topology on K with respect to {aBb [a, b~K x} as a base of zero neighbourhoods. This topology--introduced by Schr6der I-7] and Hartmann 1-2l--is a field topology which is minimal among the ring topologies of K. Let /C be the completion of K with respect to this topology. We investigate the algebraic structure of/~" and show that /~" is a ring with at most two maximal ideals. The main tool is an extension theorem of value functions on rings.
There are examples of valued skew fields which have completions with exactly two maximal ideals. Hence, minimality of the topology does not imply that the completion is a skew field.
In the first section we introduce S-topologies as a generalization of V-topologies. We show that valuation topologies are S-topologies. Since the completion of a field with a V-topology is again a field 1-31, we are mainly concerned with valuation rings which do not generate V-topologies (examples in ).
1. S-TOPOLOGIES
Consider the following conditions on a filter r of a ring R.
(A) The int~:rsection of all members of r contains only 0.
(B) For every UeT there exists Vex such that V-V_ U.