Let R be a ring. By the notation [R,R] we denote the additive subgroup of R generated by all [a,b] = ab − ba in R. In this work, we partially generalize a result due to Herstein [I. N. Herstein, Topics in Ring Theory (University of Chicago Press, 1969)] showing that if 1 ∈ [R,R], then the subring generated by [R,R] is equal to R. This result implies that [R,R] cannot be a proper subring of R.

中文翻译：

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在换向器产生的子环上

让R轴承。按符号[R,R]我们表示的加性子群R由所有人生成[一种,b] = 一种b - b一种 在R. 在这项工作中，我们部分概括了 Herstein [IN Herstein,环论专题（芝加哥大学出版社，1969 年）] 表明如果1 ∈ [R,R]，则生成的子环由[R,R]等于R. 这个结果意味着[R,R]不能是适当的子环R.