An Approximation Theorem for Coarse V-Topologies on Rings

Abstract

An approximation theorem for V-topologies on not necessarily commutative rings is proved. This holds for a certain class of rings (called rings with enough units) and a certain class of V-topologies (called coarse V-topologies). This has application, for example, to V-topologies induced by orderings.

Valuations and higher level orders in commutative rings

“…An approximation theorem for V-topologies on rings is proven in [Ma2]. As in the field and skew field cases we can apply this to the valuations we have constructed which correspond to our dependency classes.…”

“…and let τ i be the (archimedean) V-topology induced byP i . By remarks in [Ma2], each of these V-topologies is coarse. Also, they are all distinct: In the nonarchimedean case this follows from the independence of the v i 's.…”

“…By the remarks above on the V-topologies (R i , α i , τ i ), we can apply [Ma2,2.3] to our situation if we show that for each i, S * i is a τ i -neighborhood of 1. Given P ∈ O S i , let P = α −1 i (P ), i.e., P = (p i ,P ) ∈ O T .…”

“…Definition. Following [Ma2], we define a V-topology on R to be a triple (F, α, τ ) where F is a field, α : R → F a ring homomorphism such that F is the field of fractions of α(R), and τ a V-topology on F . For details, see [Ma2].…”