The study of primitivity of group rings of one relator groups

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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.