The equivalence of various generalizations of group rings and modules
The equivalence of various generalizations of group rings and modules
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DOI：
10.1007/bf01161981
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发表时间：
1982-09
影响因子：
0.8
通讯作者：
E. Dade
中科院分区：
文献类型：
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作者：
E. Dade
从我们收到的信件来看，我们关于群分级环和克利福德理论的文章 [10] 似乎引起了几位数学家的共鸣。到目前为止，这些信件中最有趣的一封来自 K.-H。 Ulbrich 指出，[10] 开头部分中的一些材料重复了他论文 [23] 的部分内容。更重要的是，他的论文和由此产生的论文 [24] 只是伽罗瓦理论的一些学生关于该主题的一系列著作中的最新著作，这些著作分别始于 Kanzaki 和 Miyashita 于 60 年代末分别发表的文章 [19] 和 [20]。因此，伽罗瓦理论家使用[10]的“强G级环”的时间至少与克利福德理论家一样长，而克利福德理论家也在六十年代末开始使用[7]。在下面的第一部分中，我们将追溯这种重复发展的一些历史，并为两组数学家研究的对象提出一个通用名称。在本说明的其余部分中，我们将解释如何使用 Miyashita 的想法将格林的顶点和源理论与当前情况联系起来。 w 1.名称和起源我们一直在研究的对象是“具有额外条件的群分级环”。通过环 9t，我们将其理解为身份为 1= 1~ 的结合环。我们修复一个恒等式 1= 1 a 的乘法群 G。一个G级环91则为一个环，也称为91，加上直和分解：
Our article [10] about group-graded rings and Clifford theory seems to have struck a chord in several mathematicians, to judge by the letters we have received. By far the most interesting of those letters came from K.-H. Ulbrich, who pointed out that some of the material in the opening sections of [10] duplicated parts of his thesis [23]. What is more, his thesis and the resulting paper [24] were only the latest in a series of works on this subject by a number of students of Galois theory, starting with the articles [19] and [20] by Kanzaki and Miyashita, respectively, at the end of the sixties. Thus the Galois-theorists had been using the'strongly G-graded rings' of [10] at least as long as the Clifford-theorists, who also started at the end of the sixties with [7]. In the first section below we shall trace some of the history of this duplicate development and suggest a common name for the objects studied by both groups of mathematicians. In the rest of this note we shall explain how an idea of Miyashita can be used to relate Green's theories of vertices and sources to the present situation. w 1. Names and OriginsThe objects we have all been studying are'group-graded rings with an extra condition'. By a ring 9t we understand an associative ring with identity 1= 1~. We fix a multiplicative group G with identity 1= 1 a. A G-graded ring 91 is then a ring, also called 91, together with a direct sum decomposition: