We introduce a theory of cyclic Kummer extensions of commutative rings for partial Galois extensions of finite groups, extending some of the well-known results of the theory of Kummer extensions of commutative rings developed by Borevich. In particular, we provide necessary and sufficient conditions to determine when a partial n-Kummerian extension is equivalent to either a radical or an I-radical extension, for some subgroup I of the cyclic group ℤn.

中文翻译：

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部分作用和循环库默理论

我们介绍了有限群的部分 Galois 扩展的交换环的循环 Kummer 扩展理论，扩展了 Borevich 开发的交换环的 Kummer 扩展理论的一些著名结果。特别是，我们提供了必要和充分条件来确定何时部分n-Kummerian 扩展等价于激进或一世-激进扩展，对于某些子组一世循环群的ℤn.