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.

中文翻译：

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带求值 II 的除环中的自由群代数

令 $R$ 是交换环 $k$ 上的代数。假设 $R$ 被赋予了在有序组 $(G,<)$ 上索引的降序过滤，使得对 $k$ 的限制是正的。我们表明，在诱导分级环 $grad(R)$ 中的某组生成元上存在自由代数意味着 $R$ 中存在自由群代数。我们最好的结果是为赋予估值的除法环获得的。