On embedding some G-filtered rings into skew fields

Abstract

We consider the filtered rings with filtration v taking values in an ordered group G (or G-filtered rings). We prove that if a ring R of this type satisfies the condition $$\forall a,b \in R^* \forall \varepsilon \in G \exists x,y \in R^* v(a \cdot x - b \cdot y) > \varepsilon \cdot v(a \cdot x)$$ then R embeds into a skew field. This skew field D becomes a topological ring in the topology induced by an extension of v, while R · R−1 is everywhere dense in D.