For a subset $T$ of a ring $R$ we denote by ${ℐ}(T)$ the ideal of $R$ generated by $T$. Given a higher commutator $L$ of $R$, if $R={ℐ}(L)$ then $R=L+L^2$? The question is motivated by the result that a ring $R$ is equal to its subring generated by $[R,R]$ if $R$ is either a noncommutative simple ring (by Herstein) or a unital ring with $1\in [R,R]$ (by Eroǧlu). In this note, we study the question for the rings $R$ satisfying the property that every proper ideal of $R$ is contained in a maximal ideal (in particular, if $R$ is finitely generated as an ideal).

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关于环的更高换向器

对于一个子集$吨$一个戒指$R$我们表示${ℐ}(吨)$的理想$R$由产生$吨$. 给定一个更高的换向器$\mathrm{大号}$的$R$， 如果$R={ℐ}(\mathrm{大号})$然后$R=\mathrm{大号}+{\mathrm{大号}}^2$? 这个问题的动机是环的结果$R$等于它的子环由$[R,R]$如果$R$是一个非交换简单环（Herstein）或一个单位环$1\in [R,R]$（由 Eroǧlu）。在这篇笔记中，我们研究了环的问题$R$满足每一个适当理想的性质$R$包含在一个极大理想中（特别是，如果$R$是作为理想有限生成的）。