We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimension. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that a pseudofinite $\widetilde {\mathfrak M}_c$ -group of finite positive dimension contains a finite-by-abelian subgroup of positive dimension, and a pseudofinite group of dimension 2 contains a soluble subgroup of dimension 2.

中文翻译：

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维度组和字段

我们将定义维度的一般概念，以及可解释集带有这种维度的研究组和环。特别是，我们推导出组的链条件，域和域的可定义性结果，并证明一个伪有限$\widetilde {\mathfrak M}_c$- 有限正维群包含一个正维有限乘阿贝尔子群，维数为 2 的伪有限群包含维数为 2 的可溶子群。