Non-noetherian groups and primitivity of their group algebras

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• 作者：James Alexander^a ; ^{jamesja@udel.edu} ; Tsunekazu Nishinaka^b ; ^{nishinaka@econ.u-hyogo.ac.jp}
• 关键词：16S34 ; 20C07 ; 20E25 ; 20E06 ; 05C15
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：473
• 期：Complete
• 页码：221-246
• 全文大小：547 K
• 卷排序：473

文摘

We prove that the group algebra KG of a group G over a field K is primitive, provided that G has a non-abelian free subgroup with the same cardinality as G, and that G   satisfies the following condition (⁎)(⁎): for each subset M of G consisting of a finite number of elements not equal to 1, and for any positive integer m, there exist distinct a, b, and c in G   so that if (x1−1g1x1)⋯(xm−1gmxm)=1, where gigi is in M   and xixi is equal to a, b, or c for all i between 1 and m  , then xi=xi+1xi=xi+1 for some i. This generalizes results of ,  and , and [18], and proves that, for every countably infinite group G   satisfying (⁎)(⁎), KG is primitive for any field K. We use this result to determine the primitivity of group algebras of one relator groups with torsion.