Free group algebras in division rings with valuation I

TL;DR

This paper investigates conditions under which free group algebras exist within division rings equipped with valuations, linking properties of graded rings to the structure of the original division rings.

Contribution

It establishes a connection between free algebras in graded rings and free group algebras in division rings with valuations, providing new criteria for their existence.

Findings

Free algebras in graded rings imply free group algebras in division rings.

Results are strongest for division rings with valuations.

Provides a method to identify free group algebras using graded ring properties.

Abstract

Let $R$ be an algebra over a commutative ring $k$. Suppose that $R$ is endowed with a descending filtration indexed on an ordered group $(G,<)$ such that the restriction to $k$ is positive. We show that the existence of free algebras on a certain set of generators in the induced graded ring $grad(R)$ implies the existence of free group algebras in $R$. Our best results are obtained for division rings endowed with a valuation.