The study of primitivity of group rings of one relator groups

本研究的目的之一是给出具有扭转的单亲缘群群环的基元性。主要方法是利用本文作者在前一篇论文中提出的sr图理论。我们可以完全解决问题，并得到比最初预期的更一般的结果。作为该结果的推论，得到了具有扭转的单关系群群环的基元性。定理设G是一个群，它有一个与G具有相同基数的自由子群。如果G满足下列条件，则群环KG对任何域K都是基元：对于G的每一个有限子集M，存在一个由G的三个元素组成的子集X，使得y_1g_1x_1…y_mg_mx_m = 1意味着对于某个i， x_i = x_{i+1}，其中g_i在M中，x_i在X中，y_i是x_i的逆。

One of our purposes of this study was to give the primitivity of group rings of one relator groups with torsion. Main method is using the SR-graph theory which is induced originally by the present author in the previous paper. We could completely solve the problem and get a more general result than initially expected. The primitivity of group rings of one relator groups with torsion is obtained as a corollary of the result. That is, we got the next theorem:Theorem Let G be a group which has a free subgroup with the same cardinality as G. If G satisfies the following condition, then the group ring KG is primitive for any field K: For each finite subset M of G, there exists a subset X which consists of three elements of G such that y_1g_1x_1...y_mg_mx_m = 1 implies x_i = x_{i+1} for some i, where g_i is in M, x_i is in X, and y_i is the inverse of x_i.